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The second part of the review is devoted to the topical areas of research aimed at creating sources of single-photon and correlated two-photon states of light, which are intended for use in long-range quantum communication systems.
Теги: long-range quantum communication systems single-photon and correlated two-photon states of light дальнодействующая квантовая связь однофотонные и коррелированные двухфотонные состояния света
The second part of the review is devoted to the topical areas of research aimed at creating sources of single-photon and correlated two-photon states of light, which are intended for use in long-range quantum communication systems.
The most important components of long-range quantum communication systems, along with quantum memory devices, are the sources of single-photon states of light. The ideal single-photon source is the source of single-photon light pulses meeting the following conditions:
• Light pulse is emitted into a target spatial-temporal mode of the electromagnetic field, i. e., the field created by the source is in a pure quantum state. In practice, it means indistinguishability of the transform-limited single-photon pulses.
• The probability of detection of the single photon at the output (efficiency of the source) is equal to 100%. It means that the output light pulse, from the one hand, should not contain the vacuum state and, from the other hand, should not contain more than one photon.
The key parameter describing the quality of the single-photon source is the fidelity, which is the measure of compliance of the output state to the target state. If the state at the output is not ideal and described by the density operator ρ, while the target one by vector , the fidelity F of the source is defined as the mean . This value equals 1 in case of full compliance, when , and equals 0 in case of maximum discrepancy (when the states are orthogonal). Experimentally, the second-order zero-time autocorrelation function is usually measured (fig. 3). Under generation in a single spatial-temporal mode of the field (stable wave packet), we obtain , where are the annihilation (creation) operators of the photons in the target spatial-temporal mode at the source output. In case of the single-photon state, this value should be equal to zero. Furthermore, in case of the single-mode field, there is one-to-one correspondence between the fidelity F and the second-order zero-time autocorrelation function, which can be written as for . A deviation of F from unit or that of from zero describes a contribution of multi-photon states into this mode. In case of multi-mode generation, the situation becomes more complicated since loss of quality can be connected not only with contribution of multi-photon states, but also with distinguishability of the photons, which corresponds to a mixed state at the source output. In this case, alone is not enough to characterize the output field. Then the suitable measure is the visibility of Hong-Ou-Mandel dip (fig. 4) reaching 100% provided only that two independent states at the input of the interferometer are single-photon and pure, i. e. indistinguishable.
Since in practice it is only possible to approximate the ideal case, it is important to understand which minimum requirements the single-photon source should meet to represent practical interest from the point of view of long-range quantum communication. As an example, let us consider theoretical estimates carried out in the review paper [64]: a quantum repeater based on single-photon sources will be more effective than others (i. e., than DLCZ scheme [65]) provided that efficiency of generation of single-photon pulses exceeds 67%, and contribution of two-photon states does not exceed 10–4. It also implies that sources create pure single-photon states. Furthermore, essential requirements are imposed by optical quantum memories which are necessary for implementation of quantum repeaters. If rare-earth ion doped crystals are used as data carriers, then quantum memory devices in the short term will be able to record and retrieve optical pulses which spectral width does not exceed several GHz. If the completely optical protocol of quantum repeater (without quantum memory) was considered [66], its implementation would require the creation of multi-photon entangled quantum states by means of huge number (of the order of one million) of single-photon sources [67].
Currently, the problem of creating an effective single-photon source is elaborated in two directions: creation of sources based on spontaneous emission of a single quantum system (quantum dots, color centers, atoms or ions in an optical trap) and creation of sources based on nonlinear optical phenomena (spontaneous parametrical down-conversion, spontaneous four-wave mixing) in extended media (crystals, waveguides, fibers). Each approach is described in detail in the review [68]. Therefore, only the basic moments are considered below and the latest experimental results are discussed.
The simplest approach to creation of single-photon sources is the use of spontaneous emission of single quantum systems initiated by a pumping pulse. The main advantages of this approach are the possibility of generation on demand and absence of contributions of two- or multi-photon states. Quantum dots [69, 70] and color centers in diamonds [71, 72], such as NV-centers and the SiV-centers, are suggested as promising quantum systems which can be used in integrated photonic circuits. When analyzing the approach in general, low collection efficiency from the point source locating, in particular, in materials with high refraction index, incoherent nature of single-photon pulses (non transform-limited pulses) due to large homogeneous broadening of optical transitions at room temperature and uniqueness of each separate center in a solid-state matrix (inhomogeneous broadening of optical transitions) thus resulting in distinguishability of the photons emitted by the different centers are specified as the main shortcomings. Furthermore, it should be also noted that spontaneously emitted single-photon pulses are of exponential fall temporal form with sharp leading edge that complicates synchronization of light pulses when implementing quantum algorithms and reduces efficiency of storage and retrieval of single-photon pulses in quantum memory devices. Today the best parameters of single-photon sources are demonstrated by quantum dots InAs/GaAs at the temperatures of liquid helium: high quality of single-photon states () [73, 74], emission of over 1000 photons with indistinguishability (the visibility of the HOM dip) exceeding 92% [75], high collection efficiency (over 65%) [73, 74, 76], generation of entangled two-photon states via cascade transitions [76, 77]. Currently, the topics of active research are electrically pumped single-photon sources, sources that are built-in or tightly connected with photonic chips, and new promising systems, such as defects in two-dimensional materials and carbon nano-tubes (see the recent review [78]).
The functioning of single-photon sources based on spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing (SFWM) is founded on the photon number correlation in the modes of the scattered field (fig. 5). Detecting one of the correlated photons in a pair (say, idler photon) unambiguously informs about existence of the second photon (signal photon). Therefore, such sources are called heralded sources. Since the actual efficiency of the heralding detector is less than 100%, the absence of the photocount does not mean the absence of the signal photon. In order to get rid of such uncontrollable contributions to the source output field, it is possible to put a gate on the way of the signal photon which is opened only on a detector signal (i. e., in the presence of a trigger pulse). The concept of such conditional preparation of single-photon states has been suggested by D. N. Klyshko in 1977 [79], and the first experiment has been performed by Hong and Mandel in 1986 [80].
The main advantages of the sources based on SPDC or SFWM are the possibility of generating photons with the broad range of frequencies, the possibility to generate pure quantum states (or tranform-limited pulses) at room temperatures, the possibility of preparation of single-photon pulses of different duration and temporal forms. The main shortcomings are the random nature of generation (during the pumping field action, the pairs of photons are born at random moments of time) and non-zero contribution of multi-photon states (along with the photon pairs, the fours, the sixs, etc. are also born).
To make the source more deterministic and simultaneously to decrease the contribution of multi-photon states, one can take advantage of multiplexing of several sources. In this case, several SPDC or SFWM processes are combined in one source so that the probability of conditional preparation of photons (heralding efficiency) could be increased without the increase in pumping power for each process separately, thus keeping high quality of the source. The development and implementation of different multiplexing schemes is an active area of research. The latest experiments on temporal [81, 82] and spatial [83, 84] multiplexing are worth noting. In particular, spatial multiplexing of two single-photon sources based on SFWM in a photonic chip is implemented in work [83]. The possibility of creating an array of identical single-photon sources in a photon chip has been recently shown in work [85].
Typical values of spectral width of bi-photon fields created in nonlinear crystals or waveguides are of several nanometers. However, in order to store and recall single photons in quantum memory devices, one needs spectral width of about 10 – 100 MHz if referred to quantum memory based on rare-earth ion doped crystals. In this respect, the use of SPDC in a resonator that allows one to reduce generation bandwidth down to tens of MHz and less, while increasing spectral brightness of the source [86–96] is very promising. Furthermore, the use of a resonator allows us to control the temporal form of single-photon pulses via modulation of the pumping pulses [97, 98] thus providing the maximum efficiency of conditional single-photon preparation. As for narrow-band single-photon sources based on SFWM, the use of ring micro-resonators [99–105] seems to be very promising. This approach allows one to solve several problems simultaneously: to increase the efficiency of generation and to reduce the required pumping power [99, 101, 105], to simplify frequency division of photons and pumping radiation filtering at the expense of large free spectral range of micro-resonators [104], to generate narrow-band photons compatible with quantum memory devices [103] and, finally, to produce scalable photonic chips [104] necessary for implementation, for example, of spatial multiplexing [83]. Furthermore, ring micro-resonators can also be useful for implementation of three-photon SPDC [106] allowing to create heralded sources of correlated photon pairs.
Turning to the general subject of this review, one can say that implementation of long-range quantum communication using quantum repeaters currently seems to be the most achievable of those ambitious tasks which are set in the field of quantum optical technologies. In order to implement the protocols of quantum repeaters, it is necessary to communicate quantum memory devices through the optical fibers. Therefore, one of the urgent tasks is the development of compatible (regarding the wavelength and spectral width) sources of non-classical states of light and storage devices allowing implementing, in one way or another, quantum communication on telecommunication wavelengths. Furthermore, both optical memories and single-photon sources have to possess high (near to 100%) efficiency. Demonstration of such devices remains the most important and complex challenge in the field of development of components of long-range quantum communication so far.
The work was supported by the Russian Science Foundation (grant No. 14–12–00806).
The most important components of long-range quantum communication systems, along with quantum memory devices, are the sources of single-photon states of light. The ideal single-photon source is the source of single-photon light pulses meeting the following conditions:
• Light pulse is emitted into a target spatial-temporal mode of the electromagnetic field, i. e., the field created by the source is in a pure quantum state. In practice, it means indistinguishability of the transform-limited single-photon pulses.
• The probability of detection of the single photon at the output (efficiency of the source) is equal to 100%. It means that the output light pulse, from the one hand, should not contain the vacuum state and, from the other hand, should not contain more than one photon.
The key parameter describing the quality of the single-photon source is the fidelity, which is the measure of compliance of the output state to the target state. If the state at the output is not ideal and described by the density operator ρ, while the target one by vector , the fidelity F of the source is defined as the mean . This value equals 1 in case of full compliance, when , and equals 0 in case of maximum discrepancy (when the states are orthogonal). Experimentally, the second-order zero-time autocorrelation function is usually measured (fig. 3). Under generation in a single spatial-temporal mode of the field (stable wave packet), we obtain , where are the annihilation (creation) operators of the photons in the target spatial-temporal mode at the source output. In case of the single-photon state, this value should be equal to zero. Furthermore, in case of the single-mode field, there is one-to-one correspondence between the fidelity F and the second-order zero-time autocorrelation function, which can be written as for . A deviation of F from unit or that of from zero describes a contribution of multi-photon states into this mode. In case of multi-mode generation, the situation becomes more complicated since loss of quality can be connected not only with contribution of multi-photon states, but also with distinguishability of the photons, which corresponds to a mixed state at the source output. In this case, alone is not enough to characterize the output field. Then the suitable measure is the visibility of Hong-Ou-Mandel dip (fig. 4) reaching 100% provided only that two independent states at the input of the interferometer are single-photon and pure, i. e. indistinguishable.
Since in practice it is only possible to approximate the ideal case, it is important to understand which minimum requirements the single-photon source should meet to represent practical interest from the point of view of long-range quantum communication. As an example, let us consider theoretical estimates carried out in the review paper [64]: a quantum repeater based on single-photon sources will be more effective than others (i. e., than DLCZ scheme [65]) provided that efficiency of generation of single-photon pulses exceeds 67%, and contribution of two-photon states does not exceed 10–4. It also implies that sources create pure single-photon states. Furthermore, essential requirements are imposed by optical quantum memories which are necessary for implementation of quantum repeaters. If rare-earth ion doped crystals are used as data carriers, then quantum memory devices in the short term will be able to record and retrieve optical pulses which spectral width does not exceed several GHz. If the completely optical protocol of quantum repeater (without quantum memory) was considered [66], its implementation would require the creation of multi-photon entangled quantum states by means of huge number (of the order of one million) of single-photon sources [67].
Currently, the problem of creating an effective single-photon source is elaborated in two directions: creation of sources based on spontaneous emission of a single quantum system (quantum dots, color centers, atoms or ions in an optical trap) and creation of sources based on nonlinear optical phenomena (spontaneous parametrical down-conversion, spontaneous four-wave mixing) in extended media (crystals, waveguides, fibers). Each approach is described in detail in the review [68]. Therefore, only the basic moments are considered below and the latest experimental results are discussed.
The simplest approach to creation of single-photon sources is the use of spontaneous emission of single quantum systems initiated by a pumping pulse. The main advantages of this approach are the possibility of generation on demand and absence of contributions of two- or multi-photon states. Quantum dots [69, 70] and color centers in diamonds [71, 72], such as NV-centers and the SiV-centers, are suggested as promising quantum systems which can be used in integrated photonic circuits. When analyzing the approach in general, low collection efficiency from the point source locating, in particular, in materials with high refraction index, incoherent nature of single-photon pulses (non transform-limited pulses) due to large homogeneous broadening of optical transitions at room temperature and uniqueness of each separate center in a solid-state matrix (inhomogeneous broadening of optical transitions) thus resulting in distinguishability of the photons emitted by the different centers are specified as the main shortcomings. Furthermore, it should be also noted that spontaneously emitted single-photon pulses are of exponential fall temporal form with sharp leading edge that complicates synchronization of light pulses when implementing quantum algorithms and reduces efficiency of storage and retrieval of single-photon pulses in quantum memory devices. Today the best parameters of single-photon sources are demonstrated by quantum dots InAs/GaAs at the temperatures of liquid helium: high quality of single-photon states () [73, 74], emission of over 1000 photons with indistinguishability (the visibility of the HOM dip) exceeding 92% [75], high collection efficiency (over 65%) [73, 74, 76], generation of entangled two-photon states via cascade transitions [76, 77]. Currently, the topics of active research are electrically pumped single-photon sources, sources that are built-in or tightly connected with photonic chips, and new promising systems, such as defects in two-dimensional materials and carbon nano-tubes (see the recent review [78]).
The functioning of single-photon sources based on spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing (SFWM) is founded on the photon number correlation in the modes of the scattered field (fig. 5). Detecting one of the correlated photons in a pair (say, idler photon) unambiguously informs about existence of the second photon (signal photon). Therefore, such sources are called heralded sources. Since the actual efficiency of the heralding detector is less than 100%, the absence of the photocount does not mean the absence of the signal photon. In order to get rid of such uncontrollable contributions to the source output field, it is possible to put a gate on the way of the signal photon which is opened only on a detector signal (i. e., in the presence of a trigger pulse). The concept of such conditional preparation of single-photon states has been suggested by D. N. Klyshko in 1977 [79], and the first experiment has been performed by Hong and Mandel in 1986 [80].
The main advantages of the sources based on SPDC or SFWM are the possibility of generating photons with the broad range of frequencies, the possibility to generate pure quantum states (or tranform-limited pulses) at room temperatures, the possibility of preparation of single-photon pulses of different duration and temporal forms. The main shortcomings are the random nature of generation (during the pumping field action, the pairs of photons are born at random moments of time) and non-zero contribution of multi-photon states (along with the photon pairs, the fours, the sixs, etc. are also born).
To make the source more deterministic and simultaneously to decrease the contribution of multi-photon states, one can take advantage of multiplexing of several sources. In this case, several SPDC or SFWM processes are combined in one source so that the probability of conditional preparation of photons (heralding efficiency) could be increased without the increase in pumping power for each process separately, thus keeping high quality of the source. The development and implementation of different multiplexing schemes is an active area of research. The latest experiments on temporal [81, 82] and spatial [83, 84] multiplexing are worth noting. In particular, spatial multiplexing of two single-photon sources based on SFWM in a photonic chip is implemented in work [83]. The possibility of creating an array of identical single-photon sources in a photon chip has been recently shown in work [85].
Typical values of spectral width of bi-photon fields created in nonlinear crystals or waveguides are of several nanometers. However, in order to store and recall single photons in quantum memory devices, one needs spectral width of about 10 – 100 MHz if referred to quantum memory based on rare-earth ion doped crystals. In this respect, the use of SPDC in a resonator that allows one to reduce generation bandwidth down to tens of MHz and less, while increasing spectral brightness of the source [86–96] is very promising. Furthermore, the use of a resonator allows us to control the temporal form of single-photon pulses via modulation of the pumping pulses [97, 98] thus providing the maximum efficiency of conditional single-photon preparation. As for narrow-band single-photon sources based on SFWM, the use of ring micro-resonators [99–105] seems to be very promising. This approach allows one to solve several problems simultaneously: to increase the efficiency of generation and to reduce the required pumping power [99, 101, 105], to simplify frequency division of photons and pumping radiation filtering at the expense of large free spectral range of micro-resonators [104], to generate narrow-band photons compatible with quantum memory devices [103] and, finally, to produce scalable photonic chips [104] necessary for implementation, for example, of spatial multiplexing [83]. Furthermore, ring micro-resonators can also be useful for implementation of three-photon SPDC [106] allowing to create heralded sources of correlated photon pairs.
Turning to the general subject of this review, one can say that implementation of long-range quantum communication using quantum repeaters currently seems to be the most achievable of those ambitious tasks which are set in the field of quantum optical technologies. In order to implement the protocols of quantum repeaters, it is necessary to communicate quantum memory devices through the optical fibers. Therefore, one of the urgent tasks is the development of compatible (regarding the wavelength and spectral width) sources of non-classical states of light and storage devices allowing implementing, in one way or another, quantum communication on telecommunication wavelengths. Furthermore, both optical memories and single-photon sources have to possess high (near to 100%) efficiency. Demonstration of such devices remains the most important and complex challenge in the field of development of components of long-range quantum communication so far.
The work was supported by the Russian Science Foundation (grant No. 14–12–00806).
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