Figure 6.1 Anti-Stokes fluorescence emission in a two-level system. Additionally, only a fraction of the absorbed photons result in excitation of coolant atoms. The rest are absorbed by background impurities. With the absorption coefficient of the coolant atoms designated by αc and that of the background by αb, the proportion of absorbed photons that contribute to cooling is h a c b( ) = + (6.3) If the assumption is made that phonon-mediated absorp-tion is proportional to the occupation probability of the phonon mode, namely n k T= (exp[ / ]-1)B -1W , the efficiency becomes h a a aabs c c b/( )= +n n . Then at high temperatures (kBT >> ħΩ)), ηabs varies little with temperature when the background absorp-tion is low (αb/αc << 1). On the other hand at low temperatures, as n Æ 0, ηabs decreases in proportion to n . Hence the absorption efficiency drops dramatically through the important cryogenic re-gime. The overall cooling is determined by the product of the various efficiency factors. h h h n n ËÁ ˆ ¯˜ (6.4) From Eqs. 6.1 and 6.4 it is easy to show that the cooling power density per unit frequency is Pc/V = ηabsηextαtotIv(vfi – v)/v. Here Iv= I0g(v) is the intensity per unit frequency and g(v) is a normalized spectral line shape function. V is the interaction volume. If the bandwidth of the light source equals the full linewidth 2Γ of the optical transition, and the density of atoms in the interaction volume is N/V, the energy loss rate per atom is R = 2ΓηcαtotIv(V/N) = ηcαtotI0(V/N) (6.5) The condition for net cooling can be expressed in a simple way based on Eq. 6.4, since ηc must exceed zero. By assuming that the laser detuning is optimized at the phonon frequency Ω, we can rewrite the numerator using hvft – hv = ħΩ. Furthermore the average phonon energy is ħΩ ≈ kBT near the boundary of the classical regime [15]. Then the condition for laser cooling (ηc > 0) becomes h h next abs B> -1 k T h (6.6) An important limitation of refrigeration based on anti-Stokes fluorescence emerges from this condition by determining the temperature at which the cooling efficiency drops to zero (ηc = 0). By solving Eq. 6.6 as an equality, it is found that a lower bound exists for the attainable temperature. The minimum temperature Tmin that can be reached with anti-Stokes cooling found from Eq. 6.6 is T h ( / ) ( ) ( / ) = + - + Ê ËÁ ˆ ¯˜ n a a h a a 1 (6.7) According to Eq. 6.7, if the quantum efficiency is high (next ≈ 1), the temperature limit for laser cooling by anti-Stokes emission is primarily determined by the frequency of light and the background-to-resonant-absorption ratio. The dependence on these two factors arises because heat input to the system is proportional to incident photon energy and absorptive heating efficiency. Only when the background absorption coefficient vanishes could arbitrarily low temperatures be reached. For the moderate levels of background absorption encountered in practice in solids, this places a severe constraint on temperatures that can be reached by laser cooling. As a simple example of this, the anti-Stokes cooling limit predicted by Eq. 6.7 at a wavelength of 1 micron is Tmin = 145.3 K when the external quantum efficiency is ηext = 0.90 and background absorption coefficient is αb = αc/1000. The lower bound on attainable temperatures can be interpreted in another way. It arises because the cooling rate depends on phonon occupation, a factor that diminishes as temperature decreases. Eventually the rate of anti-Stokes cooling drops below the rate of heating from background impurity absorption and cooling is no longer possible [16]. For 3D refrigeration of solids to be as effective as laser cooling of gases, techniques capable of maintaining high rates of cooling through the cryogenic temperature range are needed. This is discussed further in what follows. In the remainder of this chapter momentum is incorporated into the discussion to investigate whether the fundamental temperature limitation of anti-Stokes fluorescence cooling can be evaded when the dynamics are viewed comprehensively. Judging from the successes of laser cooling of gases, cooling schemes that conserve momentum and energy simultaneously during optical interactions are the most efficient. Yet momentum is disregarded in the anti-Stokes fluorescence method. Other techniques may offer the capability of balancing the momentum and energy requirements more consistently to accommodate phonon dispersion (Fig. 6.2).
Figure 6.1 Anti-Stokes fluorescence emission in a two-level system. Additionally, only a fraction of the absorbed photons result in excitation of coolant atoms. The rest are absorbed by background impurities. With the absorption coefficient of the coolant atoms designated by αc and that of the background by αb, the proportion of absorbed photons that contribute to cooling is h a c b( ) = + (6.3) If the assumption is made that phonon-mediated absorp-tion is proportional to the occupation probability of the phonon mode, namely n k T= (exp[ / ]-1)B -1W , the efficiency becomes h a a aabs c c b/( )= +n n . Then at high temperatures (kBT >> ħΩ)), ηabs varies little with temperature when the background absorp-tion is low (αb/αc << 1). On the other hand at low temperatures, as n Æ 0, ηabs decreases in proportion to n . Hence the absorption efficiency drops dramatically through the important cryogenic re-gime. The overall cooling is determined by the product of the various efficiency factors. h h h n n ËÁ ˆ ¯˜ (6.4) From Eqs. 6.1 and 6.4 it is easy to show that the cooling power density per unit frequency is Pc/V = ηabsηextαtotIv(vfi – v)/v. Here Iv= I0g(v) is the intensity per unit frequency and g(v) is a normalized spectral line shape function. V is the interaction volume. If the bandwidth of the light source equals the full linewidth 2Γ of the optical transition, and the density of atoms in the interaction volume is N/V, the energy loss rate per atom is R = 2ΓηcαtotIv(V/N) = ηcαtotI0(V/N) (6.5) The condition for net cooling can be expressed in a simple way based on Eq. 6.4, since ηc must exceed zero. By assuming that the laser detuning is optimized at the phonon frequency Ω, we can rewrite the numerator using hvft – hv = ħΩ. Furthermore the average phonon energy is ħΩ ≈ kBT near the boundary of the classical regime [15]. Then the condition for laser cooling (ηc > 0) becomes h h next abs B> -1 k T h (6.6) An important limitation of refrigeration based on anti-Stokes fluorescence emerges from this condition by determining the temperature at which the cooling efficiency drops to zero (ηc = 0). By solving Eq. 6.6 as an equality, it is found that a lower bound exists for the attainable temperature. The minimum temperature Tmin that can be reached with anti-Stokes cooling found from Eq. 6.6 is T h ( / ) ( ) ( / ) = + - + Ê ËÁ ˆ ¯˜ n a a h a a 1 (6.7) According to Eq. 6.7, if the quantum efficiency is high (next ≈ 1), the temperature limit for laser cooling by anti-Stokes emission is primarily determined by the frequency of light and the background-to-resonant-absorption ratio. The dependence on these two factors arises because heat input to the system is proportional to incident photon energy and absorptive heating efficiency. Only when the background absorption coefficient vanishes could arbitrarily low temperatures be reached. For the moderate levels of background absorption encountered in practice in solids, this places a severe constraint on temperatures that can be reached by laser cooling. As a simple example of this, the anti-Stokes cooling limit predicted by Eq. 6.7 at a wavelength of 1 micron is Tmin = 145.3 K when the external quantum efficiency is ηext = 0.90 and background absorption coefficient is αb = αc/1000. The lower bound on attainable temperatures can be interpreted in another way. It arises because the cooling rate depends on phonon occupation, a factor that diminishes as temperature decreases. Eventually the rate of anti-Stokes cooling drops below the rate of heating from background impurity absorption and cooling is no longer possible [16]. For 3D refrigeration of solids to be as effective as laser cooling of gases, techniques capable of maintaining high rates of cooling through the cryogenic temperature range are needed. This is discussed further in what follows. In the remainder of this chapter momentum is incorporated into the discussion to investigate whether the fundamental temperature limitation of anti-Stokes fluorescence cooling can be evaded when the dynamics are viewed comprehensively. Judging from the successes of laser cooling of gases, cooling schemes that conserve momentum and energy simultaneously during optical interactions are the most efficient. Yet momentum is disregarded in the anti-Stokes fluorescence method. Other techniques may offer the capability of balancing the momentum and energy requirements more consistently to accommodate phonon dispersion (Fig. 6.2).
To overcome the ETU-induced heating and realize efficient and enhanced laser cooling in Ho3+-doped solids, in Section 4.3, we will provide a modified excitation scheme based on ground-state absorption (GSA) of 5I8 Æ 5I7 at around 2.07 μm and ESA of 5I5 Æ 5F5 at around 2.28 μm. Compared with the conventional pumping scheme, the co-pumping scheme ameliorates the cooling performance significantly through alleviating the upconversion-quenching-induced heat [12], that is, the heat-producing 5I5 Ho3+ions are pumped to the 5F5 state by the laser at 2.28 μm. Fortunately, the 5F5 Ho3+ ions barely transit to the lower nearby state and tend to decay radiatively. Therefore, major heat produced in the conventional laser cooling of Ho3+-doped solids is avoided, and both cooling efficiency and cooling power density can be improved. In Section 4.4, we broaden the research domain of optical refrigeration to rare earth ion co-doped systems. Specifically, when Ho3+–Tm3+ co-doped YLiF4 crystals are pumped at around 2.07 μm, Ho3+ ions will populate at the 5I7 manifold. Meanwhile, Tm3+ ions will also be sensitized by absorbing energy transferred from 5I7 Ho3+ ions. Since the average energy of fluorescence photons generated by Tm3+3F4 Æ 3H6 radiative transitions is higher than that generated by Ho3+5I7 Æ 5I8 radiative transitions, energy transfer (ET) from Ho3+ to Tm3+ ions will induce annihilation of host phonons, and bring enhancement in both the cooling efficiency and the cooling power. Therefore, this cooling mechanism is referred next as energy transfer-enhanced laser cooling (ETLC) of solids [11]. 4.2 Conventional Laser Cooling of Holmium- The main upconversion process and the anti-Stokes fluorescence cooling (ASFC) processes of Ho3+-doped fluoride crystals are illustrated in Fig. 4.1. The ground-state (5I8) Ho3+ ions are pumped to the first excited state (5I7) with laser excitations of energy hν (solid red arrow on the left) and then relax thermally among the Stark levels in a timescale of picoseconds with host phonon absorption. Part of the Ho3+5I7 ions de-excite by emitting fluorescence photons of mean energy hν10 (solid blue arrow, ν10 > ν), which accounts for the ASFC process. The rest of the 5I7 Ho3+ ions de-excited and transfer their energy to the neighboring 5I7 Ho3+ ions, causing population accumulation at the third excited state (5I5) with host phonon absorption (solid green arrows). Comparison between the energy mismatch and the host phonon energy (for YLiF4 crystal host, ħω ≈ 450 cm-1) indicates that about two host phonons are annihilated for each ETU process (see Table 4.1).
Our choice of rubidium atoms as the optically active species becomes now clear. The large elasticity of collisions between the electronically excited states of the rubidium D-lines and noble gas atoms has long been proven [25]. The quasi-molecular potential has a depth of the order of kBT, allowing for a significant part of the atoms’ kinetic energy to be extracted. Finally, the cooling transitions are in the near-infrared spectral regime, making them conveniently optically accessible to commercial laser systems. The cooling power Pcool from these experiments can be estimated in a similar way as for anti-Stokes cooling [26, 27] from the mean fluorescence frequency νfl, the incident power Popt, and the absorption probability of the gas mixture at incident frequency ν, α(ν).P Pcool opt fl= -a n n nn( ) (1.1) The typical energy that can be extracted from the sample per cooling cycle is of order of the thermal energy kBT. In the experiments performed so far, we observe cooling powers Pcool in the order of up to 100 mW [17, 23, 28, 29]. Note that it is important to choose a detuning with a high-enough absorption probability a(ν). We are aware that the collisional broadening plays a crucial role here. A measurement of typical fluorescence data recorded with the dense gas mixture will be shown in Section 1.4.1. Experimentally, it turns out to be crucial that the used buffer gas is of high purity to suppress the influence of concurring heating effects, of which quenching is the most known. In the case of quenching, the excitation energy is transferred to an atom of a third species polluting the dense gas mixture when colliding with the quasi-molecule. Relaxation can then occur via nonradiative decay, and so energy is kept in the system, heating the gas. To reduce possible quenching, an effect known to occur, for example, from an admixture with nitrogen, we use a buffer gas with high purity, namely argon gas preferably with a purity of 99.9999 (argon 6.0). Residual heating can also be caused by energy pooling. In this scenario, during the collision of two excited alkali atoms one of them gets further excited. Its relaxation to the ground state implicates the emission of a fluorescence photon with higher energy. For the case of rubidium atoms, the reaction equation of that process can be written as [30]: Rb Rb Rb Rb* * **+ + + Æ + + +mv mv mv mv12 22 12 22 2 2 2 2 . (1.2) The sum of the energies of the single excited rubidium atoms before the collision is higher than the energy of the twice excited atom. The surplus of the energy is thus kept in the system as kinetic energy of the collisional partners [31]. For rubidium atoms subject to a high-pressure noble gas, fluorescence originating from the 6p Æ 5s transition then occurs, and can be detected as blue light at a wavelength λ ≈ 420 nm in the experiments. At room temperature, the alkali atoms are in the solid phase. To produce a sufficient alkali vapor number density the system is heated to temperatures of up to 680 K, while the buffer gas pressure ranges between 50 bar and 230 bar. Under these conditions, we calculate that the dense gas ensemble consists of approximately 1017 cm-3 alkali atoms and 1021 cm-3 noble gas atoms. Then, we can estimate the mean free path length λfp for a gas of colliding atoms (cross section σ) and number density n l sfp = n (1.3) Let us consider the geometric cross section πd2, where d denotes the diameter of a rare gas atom, which is twice the van der Waals radius (d = 2rvdW = 3.76 × 10−10 m for argon [32]). Then, for argon pressure of 200 bar, the mean free path length is in the order of only a few nanometers, λfp ≈ 2.3 nm. Following the Boltzmann distribution, still considering a temperature T = 680 K, one can also estimate the collisional rate to 2.4 × 1011/s. Comparing this to the natural lifetime of the first excited alkali atom level (27 ns), we calculate that 104collisions occur within this period of time. 1.3 Experimental Setup and MethodsIn the following section, we present our experimental setup in detail, including the techniques used to measure atomic fluorescence and temperature changes inside the gas mixture. We offer insights into the high-pressure cells employed (Section 1.3.1), as well as into the spectroscopic techniques (Section 1.3.2) and the thermal deflection setup (Section 1.3.3).
Publisher Summary This chapter focuses on the concept of laser cooling of a solid, which may occur when the average energy of the photons emitted by the solid is larger than the energy of the ones it absorbs. More formally, the anti-Stokes emission, which occurs at frequencies larger than that of the pump laser, must dominate the Stokes emission that occurs at smaller frequencies. A crucial additional requirement is that the nonradiative decay rates of the laser-pumped states be negligible in comparison to their radiative decay rates. For the sake of specificity, these states will be taken to be those of a set of isolated ions embedded within an insulating host, although in general they could equally well be those of gas-, liquid-, or solid-phase neutral atoms or molecules, or even those of the energy bands of intrinsic semiconductors. Outside the realm of impurity-doped insulators, recent experiments with semiconductors make net anti-Stokes fluorescence cooling of these materials appear achievable. A long-term idea for improving laser cooling of solids takes advantage of photonic band gap materials. The idea is to fabricate a refractive-index structure within the cooling element that would establish a three-dimensional photonic gap immediately below the pump frequency. In this way, emission at Stokes frequencies falling within the gap would be inhibited, thereby enhancing the cooling power derived from the anti-Stokes emission.
Expectedly there are challenging issues to be tackled. For example, the purity level of the (undoped) fiber needs to be high to avoid excessive parasitic absorption of photons by impurities. When the fiber cooler and the object to be cooled are macroscopic (centimeter scale or larger), managing the (conductive and convective) heat and photon flows is also a pertinent issue. These and related issues will be discussed below. In Section 3.2, we describe our theoretical developments in this investigation. We discuss the calculation of the cooling powers of the Tm atoms and of a model isolated fiber and the simulation of heat flow in the cooling of the heat load. In Section 3.3, we discuss experimental progress in fabricating high-purity fiber and temperature measurements under optical pumping conditions. 3.2 Theoretical Developments 3.2.1 Modeling the Cooling of Thulium AtomsThe basic cooling agents in our setup are the thulium (Tm) atoms doping the fiber. We describe in this subsection a photoexcitation model of a Tm atom in a glass host, which we use to estimate its laser cooling efficiency. More detail can be found in [7]. The model, given in Ref. [8], is shown in Fig. 3.2. It consists of four energy levels, each with a degree of degeneracy gi, as designated in the figure. The lower two levels represent the manifold of states created by the splitting of the 13-fold degenerate ground state (3H6) of Tm by interactions with the glass host, and the upper two levels represent the split ninefold excited-state (3F4) manifold. The pump laser energy hn is tuned to the energy between |1> and |2>, which is the lowest excitation energy between the two manifolds. The pumped Tm atoms de-excite (incoherently) to a level in the ground-state manifold via luminescence, at rate Wrad, and nonradiative transitions, at a rate Wnr. If the de-excitation happens sufficiently slowly, the excited atoms have time to thermalize in the excited-state manifold. The luminescent photons are thus at higher energies than that of the pump photons. In each cycle of operation in which a Tm atom absorbs a pump photon and emits a luminescent photon, which then escapes, heat is removed from the system. The population dynamics of the four-level model obeys the following rate equations [8]: ( ) g B/inc1 1 112 1 2 2 3 1 1 02 02 E k TIdN g gRN N N N w N N edt g h g ds n -Ê ˆÊ ˆ= - - + + - -Á ˜Á ˜Ë ¯ Ë ¯ (3.1) u B/inc 32 1 dN dt RN w N g 2 2= - - - Ê ËÁ ˆ ¯˜ -d u B/ (3.3) N N N N Ntotal = + + +0 1 32 (3.4)where Ni and gi are the population density and degree of degeneracy, respectively, of level |i>, and Ntotal is the total density of Tm atoms. The pump laser has intensity Iinc and frequency n. R = 2Wrad + 2Wnr, and s12 denotes the photoabsorption cross section of the Tm atoms. dEg and w1 are the energy difference between the two levels and the intramanifold relaxation rate in the ground-state manifold, and dEu and w2 are the corresponding quantities for the excited-state manifold. T is the sample temperature, and kB is the Boltzmann constant. In Eq. 3.1, the first term on the right side represents stimulated absorption from and emission into the pump laser mode. The second term models the decay of the excited manifold population and the third term the thermalization within a manifold. The terms in Eqs. 3.2 and 3.3 have the same interpretation.
The material metrics for optimal laser cooling of ion-doped solids are derived using atomic and molecular dynamics properties of the constituents. The anti-Stokes process is modeled as an optical phonon coupling of the bound electron, followed by a photon absorption. The transition dipole moment is estimated using a simplified charge-displacement model and the Judd--Ofelt theory of rare-earth ions both suggesting that transitions with high-energy gaps and similar angular momentum states should be used. The electron-phonon coupling is interpreted as a derivative of electronic energy with respect to displacement of the nearest neighboring ligands whose stretching mode frequency is approximated using the molecular data. The Debye-Gaussian model is used for the phonon density of states of diatomic crystal. Then, the Fermi golden rule is used for photon-induced, phonon-assisted electronic transition probability and applied to the cooling rate equation by defining a phonon-assisted transition dipole moment. Based on the material metrics, an example blend is investigated for its cooling performance and a general guide is proposed for selection of better performing laser cooling hosts. Furthermore, the cooling rate limits are discussed and three distinct characteristic times are identified with the photon-induced, phonon-assisted transition time controlling the rate. The metrics guide the selection of host materials for optimal cooling, and predict a noticeable increase in the absorption rate when using a blend of cation atoms.
The usual form of electromagnetic forces is given by but for neutral atoms q = 0. The next order of force is the dipole term, but this also vanishes because neutral atoms have no inherent dipole moment. However, a dipole moment can be induced by a field, and this is most efficient if the field is alternating near the atomic resonance frequency. Since these frequencies are typically in the optical range, dipole moments are efficiently induced by shining nearly resonant light on the atoms. Laser cooling and trapping rely on the interaction between laser light and atoms to exert a controllable force on the atoms, and many sophisticated schemes have been developed using the special properties of the interaction. This chapter begins with the simplest schemes for exerting optical forces on atoms, namely a single-frequency light field interacting with a two-level atom. Although this is the simplest possible scheme, it is pedagogically valuable because it shows many of the features that will be encountered further on. The ultimate temperature using such a scheme is limited, but lower temperatures can be achieved with multilevel atoms. This topic is addressed later in the chapter. Two kinds of optical forces When an atom absorbs nearly resonant light, it makes a transition to the excited state, and its subsequent return to the ground state can be either by spontaneous or by stimulated emission. The nature of the optical force that arises from these two different processes is quite different, and will be described separately [1]. The spontaneous emission case is different from the familiar quantum mechanical calculations using state vectors to describe the state of the system, because spontaneous emission causes the state of the system to evolve from a pure state into a mixed state (see Exercise 6.9). Spontaneous emission is an essential ingredient for the dissipative nature of the optical forces in monochromatic light, and the density matrix is needed to describe it (see Chap. 6). When an atom absorbs light, its energy excites the atom and its angular momentum changes the electron's orbit. Both of these are internal atomic properties. However, the linear momentum of the light, can change only the external, translational motion of the atom.