Quantum refrigerator driven by nonclassical light

We study a three-level quantum refrigerator which is driven by a generic light state, even a nonclassical one. With the help of P function expansion of the driving light, we obtain the heat current generated by different types of light states. It turns out all different input light states give the same coefficient of performance for this refrigerator, while the cooling power depend not only on the light intensity but also the specific photon statistics of the driving light. Comparing with the coherent light with the same intensity, the driving light with super(sub)-Poissonian photon statistics could raise a smaller (stronger) cooling power. We find that this is because the bunching photons would first excite the system but then successively induce the stimulated emission, which draws the refrigerator back to the starting state of the cooling process and thus decreases the cooling current generation. This mechanism provides a more delicate control method via the high order coherence of the input light.


I. INTRODUCTION
When a quantum heat engine runs between two reservoirs containing specific quantum coherences, the efficiency of the heat engine could exceed the Carnot limit between two canonical thermal baths [1][2][3][4].But such exotic effects are restricted for practical applications since quantum coherences are usually quite fragile confronting the noises surrounded.In contrast, applying a quantum light to control or drive a quantum refrigerator is feasible under current techniques, promising intriguing properties.Some recent studies show that, comparing with the normal laser light with the same intensity, using nonclassical squeezed light could help enhance the two-photon absorption rate [5], and exceed the cooling limit in laser cooling experiments [6,7].
Therefore, here we study a quantum refrigerator which is driven by different types of light states, especially, the nonclassical lights.We focus on a typical quantum thermal machine composed of a three-level system in contact with two heat baths [8].Applying a proper temperature difference to the two baths, a population inversion is generated between two levels, and the system could work as a heat engine, emitting laser light as its output work [9][10][11][12].Reversely, when a driving light is input to the three-level system, it works as a refrigerator [Fig.1(a)], moving the heat from the cold bath to the hot one [13].It is also worth noting that the transition structure of this three-level refrigerator is analogous to many other physical systems, such as the sideband cooling system [Fig.1(c)] [14], and photovoltaic systems [15,16].
When the driving light shining on the refrigerator is a generic quantum state, it is no longer enough to treat the driving light simply as a planar wave, which is a quasi- * lishengwen@bit.edu.cnclassical description in literatures.To study the interaction with a nonclassical driving light, notice that, with the help of the P function representation, a generic light state can be regarded as the combination of many coherent states |α with P (α, α * ) as the "quasi-probability", while the coherent states |α are the quantum correspondences for the classical planar waves [17][18][19][20].Therefore, the full system dynamics can be obtained as the P function average of many evolution "branches", and each evolution branch can be obtained from the above quasiclassical approach, treating the driving light as a planar wave [21][22][23][24].
Based on this approach, we obtain the cooling power of this quantum refrigerator for different input light states.It turns out the coefficient of performance (COP) always remains as e = ω c /(ω h −ω c ) ≤ T c /(T h −T c ), whose upper bound is just the Carnot limit for refrigerators.But the cooling powers generated by different driving lights depend on the specific photon statistics.Comparing with the coherent light with the same intensity, the driving light with super(sub)-Poissonian photon statistics would raise a smaller (stronger) cooling power.
We find that, this is because bunching photons would block the cooling current generation due to the stimulated emission they bring in.When a pair of bunching photons income together, the first photon would excite the refrigerator system up, but the second photon successively followed would induce the stimulated emission, drawing the system back to the previous state, and that blocks the generation of the cooling current flowing to the hot bath.Therefore, comparing with the coherent light where the photons income randomly both in bunches and individually, the bunching (antibunching) light could generate a smaller (stronger) cooling power under the same light intensity.Clearly, the similar mechanism could also take effect in many other systems undergoing light driving.
As a comparison, we also consider the situation that the whole multimode light field is in the thermal equilibrium state with a temperature T e [25][26][27][28].It turns out the thermal photon number must be larger than a certain threshold so as to make sure the system works as a refrigerator.Again that indicates the working status of the system is determined not only by the light intensity but also the specific quantum state of the light field.
The paper is arranged as follows.In Sec.II we show the basic properties of the three-level system quantum refrigerator under a coherent driving light.In Sec.III we discuss the situation that the driving light carries a generic photon statistics.In Sec.IV we show that the driving light with antibunching statistics could enhance the cooling power.In Sec.V we consider the situation that the whole multimode EM field is in a thermal equilibrium state.The summary is drawn in Sec.VI, and some detailed derivations are placed in the Appendices.

II. THE THREE-LEVEL QUANTUM REFRIGERATOR
The basic setup of the three-level quantum refrigerator is shown in Fig. 1 ] † as the transition operator.We consider the two heat baths stay in the thermal equilibrium states with the temperatures T h > T c .It is worth noting that indeed the basic transition structure of the sideband cooling system is quite similar as this three-level model [Fig.1(c)].
Here we show that, when using a light beam to drive the transition |e 1 ↔ |e 2 , such a three level system could work as a quantum refrigerator, namely, the net energy flux would flow from the cold bath to the hot one [13,28].
Generally, the driving light is modeled as a classical planar wave with a single frequency mode, and its interaction with the three level system is described by [17][18][19][20] ( where ω d is the driving frequency, d := ℘ σ− + h.c. is the dipole operator of the three level system, with ℘ := e 1 | d|e 2 as the transition dipole moment, and σ− := |e 1 e 2 | := (σ + ) † .
Under the Born-Markovian-rotating-wave approximation [29], the dynamics of the system can be described T X T I g l L w n 5 + Q z + e J 9 8 r 5 6 3 7 z v M 6 r X u e h 5 R F r h / f w L / H k N I A = = < / l a t e x i t > |e 2 i < l a t e x i t s h a 1 _ b a s e 6 4 = "       by the following master equation1 where ∆ := Ω − ω d is the detuning between the driving light and the transition frequency Ω := e 2 − e 1 , E := e iφ d ( ℘• E d )/2 is the driving strength, and the dissipations terms are gives the spontaneous emission to the EM field.
From the master equation ( 2), the changing rate of the system energy gives where Q e , Q c , Q h are the energy flux flowing to the system from the EM field, cold and hot baths respectively.In the steady state ∂ t Ĥs = 0, the above energy flows can be obtained by solving master equation (2) (Appendix B).Here we consider the situation that the spontaneous emission rate κ is negligible comparing with the coupling strengths with the two heat bath (γ h,c κ → 0), and that gives the heat flows as (let where J is the population flux, Notice that, as long as nc − nh ≥ 0 with E = 0, we have Q c ≥ 0 and Q h ≤ 0, which means the heat is flowing across the system from the cold bath to the hot one.Namely, the incoming light is driving the system to work as a refrigerator, and the above inequality gives the cooling condition as ω c /T c ≤ ω h /T h .When the driving strength is weak, the cooling power is proportional to the light intensity As a result, the coefficient of performance (COP) for this three-level refrigerator gives Therefore, this COP is just bounded by the Carnot limit for refrigerators.The equality holds when the energy flows [Eq.( 5)] approach zero, which indicates the quasistatic and reversible process, leading to the zero power.
When the spontaneous emission rate κ is a small but finite value, the upper bound of the COP would be smaller than the Carnot limit.

III. THE DRIVING LIGHT WITH GENERIC PHOTON STATISTICS
Now we consider a more general situation that the input light driving the refrigerator could carry different kinds of photon statistics, but still has a quite small linewidth and can be regarded as a monochromatic light.
In this situation, it is no longer enough to treat the driving light only as a classical planar wave, which cannot reflect the photon statistics of the input light.Here we consider the EM field is fully quantized, which is described the multimode Hamiltonian Ĥe = ω k â † kς âkς .The driving mode stays in a specific quantum state, while all the other field modes are in the vacuum state.Generally, the quantum state of the driving mode always can be represented as the following P function, Formally, this density state ˆ d of the driving mode can be regarded as the combination of many coherent state |α , with P (α, α * ) as the quasi-probability.For the quantized EM field, a single mode coherent state |α corresponds to a classical planar wave, since the electric field operator Ê(x, t) ≡ Ê− + Ê+ gives where Ê± are the field operators with positive and negative frequencies, and In this sense, the generic state ˆ d of the driving light could be regarded as a "probabilistic" combination of many classical planar waves.Therefore, the system dynamics also could be obtained as the combination of many evolution "branches", and in each branch the system is driven by the planar wave given by Eq. ( 9), that is (a rigorous proof is shown in Appendix A), where the master equation for ρ(α) (t) has the same form as the above Eq.( 2) for the quasi-classical driving [21][22][23][24].Here the driving light in Ṽα should be replaced by the planar wave in Eq. ( 9) determined by the coherent state |α , and the driving strength in Eq. ( 2) now should be modified as E α := e iφα ( ℘ • E α )/2 respectively.Correspondingly, the expectations of the system observables are obtained as where ô(α) Namely, the full system evolution ôs (t) could be regarded as the "probabilistic" summation of many evolution "branches" ô(α) s (t) , with P (α, α * ) as their probabilities, and each "branches" ô(α) s (t) can be obtained from the above master equation with the quasi-classical driving.
Based on this method, now we study the heat flows of the above quantum refrigerator when the driving light carries generic photon statistics.Similar as the energyflow conservation relation (4), the three heat flows are given as Q h = − ω h J, Q c = ω c J and Q e = Ω J, where J is the population flux obtained from the P function average (11), namely, Here J α is the same as the above Eq.( 5), which indicates the steady state flux when the driving light is a classical planar wave corresponding to the coherent state as Eq. ( 9).As a result, in spite of the driving light state, the COP for this refrigerator is still e = ω c /(ω h − ω c ) as long as J ≥ 0, which would lead to the same cooling condition as Eq. ( 7).

IV. ENHANCING THE COOLING POWER BY ANTI-BUNCHING PHOTONS
As long as the driving light state P (α, α * ) is known, the heat flows of the refrigerator can be obtained from the above Eq.( 12).Here we consider some typical examples of different driving light.
We first consider the driving light is carrying the thermal statistics, whose P function is given by P th (α) = (πn th ) −1 exp[−|α| 2 /n th ], with nth as the mean photon number.Such a distribution is also consistent with the classical picture for the chaotic light, which is regarded as the probabilistic combination of planar waves, whose intensities satisfy the negative exponential distribution [30,31].In this case, from Eq. ( 12), the population flow in the steady state gives Here ξ 0 := ℘ d 2 ω d / 0 V is the single photon coupling strength, where V takes the coherence volume of the driving light [30].And the intensity of the driving light (the Poynting vector) is It turns out the monochromatic "thermal" light is also driving the system to work as a refrigerator rather than warming it up.But comparing with the coherent driving [Eq.( 5)] under the same light intensity, the driving light with thermal statistics generates a smaller cooling power [see Fig. 2(c)], while the COP keeps the same as Eq.(7).
A more attracting situation is when the driving light has nonclassical photon statistics, e.g., the antibunching light.For nonclassical light states, their P functions may contain negative parts, or could be highly singular functions, thus cannot be regarded as legal probability distributions [17][18][19][20].In these cases, besides adopting the specific P function directly, the above average (12) also can be calculated in the following way: since the P function average is equal to the normal order expectation by making the replacement α, α * → â, â † , | E α | 2 → Ê− Ê+ [32,33], the above population flow becomes [20,24,34] Here : Ĵ(â † , â) : denotes the normal order expectation.Thus, once the generation function F (s) ≡ : exp(−sâ † â) : is obtained for the driving light statistics, the heat flows of the refrigerator can be calculated from the above integral.
Here we consider two examples of the photon statistics for the driving light, i.e., with Z ± (λ) as the normalization factors (see Appendix C).By checking the mean photon number n and variance δn 2 , we can verify P The cooling currents generated by these two types of photon statistics are shown in Fig. 2(c).Comparing with the coherent light with the same intensity, the driving light with super(sub)-Poissonian photon statistics produces a smaller (stronger) cooling power; with the increase of the driving light intensity, the cooling powers generated by the super-and sub-Poissonian lights both converge to the one generated by the coherent light.In contrast, the cooling power generated by the thermal light always keeps a finite difference lower than the coherent light situation.
This result can be explained with the help of the following expansion for the above normal order expectation (14), Here is the k-order coherence function of the driving light (zero time), and I E ≡ 2c 0 Ê− Ê+ is the light intensity.
It turns out, when the light intensity is not too strong, the heat flow induced by the driving light is proportional to the light intensity I E in spite of the photon statistics [see the first term of Eq. ( 16)].With the increase of the light intensity I E , the high order coherence g (k) (k ≥ 2) of the driving light also take effect to the heat flow.From the minus sign of the second term in Eq. ( 16), comparing with the situation of coherent driving g (2) = 1, the  ! " < l a t e x i t s h a 1 _ b a s e 6 4 = " N y w y Y 4 g a w l v a q h 2 l N 8  12).Here we set ξ0/γ = 1.1 and nc = 0.5, nh = 1.(d) Demonstration for the heat current blockage by a pair of bunching photons.When two photons income together, the first photon would excite the system up, but the second photon successively followed would induce the stimulated emission, which draws down the energy back before it flows to the hot bath, and that decreases the cooling power.
Moreover, it is worth noting that, for the two types of photon statistics (15), they both give g (2) sub/super → 1 when the light intensities grow large.Therefore, the cooling powers they generate converge to the coherent light situation.In contrast, the thermal light always gives g (2) th = 2 in spite of the light intensity, thus the cooling power generated by the thermal light always keeps a finite difference lower than the coherent light situation.
This effect can be understood by the demonstration in Fig. 2(d).Generally, one incoming photon would excite the system up |e 1 → |e 2 , and then generate an energy flow to the hot bath.But if a pair of bunching photons come together, after the system is excited to |e 2 by the first photon, the second photon successively followed could immediately induce the simulated emission and draw back the system to |e 1 , which prevents the energy flowing to the hot bath.Such a blockage effect depends on the competition between the releasing rate to the hot bath (∼ γ h ) and the stimulated emission rate due to the incoming photons (∼ ξ 0 ).
For a driving light with the Poissonian statistics, the photons income to the system randomly, either in bunches or individually, while the bunching photon pairs would decrease the cooling current flowing from the cold bath to the hot one.Therefore, comparing with the coherent light with the Poissonian statistics, the bunching (antibunching) light, which has the super(sub)-Poissonian statistics, would contribute more (less) blocking effect due to the stimulated emission, and thus produces a smaller (stronger) cooling power.It should be emphasized that here we focus on the situation that the linewidth of the driving light is negligible comparing with the decay rates γ h,c , thus the corrections from the finite bandwidth of the driving light are omitted.

V. THE WHOLE LIGHT FIELD AS A THERMAL BATH
As a comparison, here we consider another situation that the multimode EM field as a whole is a heat bath [25][26][27][28], staying in the thermal equilibrium state ρ e ∝ exp[− Ĥe /k b T e ] (T e is the temperature), and there is no other driving light beam [see Fig. 1(b)].
In this case, the incoherent thermal lights are injecting to the system from all different directions with different frequencies.The system dynamics is now described by the following master equation [29], where ne := [exp( Ω/k b T e ) − 1] −1 is the mean thermal photon number, and L h,c [ρ] are the same as Eq. ( 3) indicating the dissipation due to the coupling with the hot and cold baths.Similarly as Eq. ( 5), the energy flows from system to the hot, cold, EM baths are gives by the above master equation, i.e., ∂ t Ĥs = Q e + Q c + Q h .In the steady state, they give and N is the same as Eq. ( 6).To make the system work as refrigerator, namely, the heat flows from the cold bath to the hot one, the above heat flows require J > 0, and that gives by substituting the Planck functions.When this condition is not satisfied, the heat flows from the hot bath to the cold one, and the system is not working as a refrigerator.Since ω h − ω c ≡ Ω, this cooling condition requires that the three temperatures must satisfy T c < T h ≤ T e .
Clearly, the cooling condition here is different from the situation that the refrigerator is driven by a monochromatic light beam, which only requires nc − nh ≥ 0 and nonzero intensity for the driving light [Eq.( 5)].Unlike the above monochromatic driving case, here the incoherent thermal lights are coming to the system from all different directions with different frequencies.It turns out the cooling condition here requires that the mean thermal photon number ne (Ω, T e ) in the EM bath must be larger than a certain threshold ne ≥ nh (n c + 1)/(n c − nh ).And the COP of this refrigerator is whose upper bound is smaller than the above monochromatic driving case [Eq.( 7)].Such an upper bound also has been obtained in some previous studies about the quantum absorption refrigerators working between three thermal baths [25][26][27].Clearly, the working status of the refrigerator is significantly dependent on the the specific quantum state of the EM field, not only on the incoming light intensity.

VI. DISCUSSION
In the paper, we study a three-level quantum refrigerator which is driven by a generic light state, even a nonclassical one.Since the driving light input to the refrigerator could be a generic quantum state, it is no longer enough to treat the driving light simply as a planar wave, which is a quasi-classical description in literatures.With the help of the P function representation, a generic driving state can be regarded as the combination of many coherent states |α with P (α, α * ) as the "quasi-probability", while the coherent input states could well return the quasi-classical driving description.Therefore, the full system dynamics can be obtained as the P function average of many evolution "branches", and each evolution branch is obtained from the quasi-classical approach by treating the driving light as a planar wave.
Based on this approach, it turns out all different input light states give the same COP for this refrigerator, while the cooling power depend not only on the light intensity but also the specific photon statistics of the driving light.Comparing with the coherent light with the same intensity, the driving light with super(sub)-Poissonian photon statistics could raise a smaller (stronger) cooling power.We find that this is because the bunching photons could block the cooling current generation due to the the spontaneous emission they enhanced.This mechanism could provide a more delicate control method via the high order coherence of the input light.
As a comparison, we also consider the situation that the multimode EM field as a whole is in the thermal equilibrium state, and the incoherent thermal lights with different frequencies are injecting to the system from all different directions.It turns out the thermal photon number in the EM bath must be larger than a certain threshold so as to make the system work as a refrigerator.Therefore, the working status of the refrigerator is significantly dependent on the the photon statistics and frequency distribution of the EM field, not only on the incoming light intensity.and the transition frequency Ω ≡ e 2 − e 1 , under the interaction picture defined by Ĥ∆ := ĤS − ∆|e 2 e 2 |, the expectations of n1( 2 (B3) Based on the master equation, the heat flows defined in Eq. ( 4) give , By substituting the steady state solutions (B2) into the above flows, they give the steady heat flows as Eq. ( 5) in the main text.
Appendix C: The normal order expectation for the heat flow generated by generic photon statistics Here we show how to calculate the normal order expectation for the population flow in Eq. ( 14).Notice that the population flow has been turned into the integral of the normal order expectation : e −sâ † â : ≡ F (s), which is the critical part to be calculated.Here we consider the density state of the monochromatic driving light is diagonal in the Fock basis, i.e., ρ = P n |n n| with P n as the photon number distribution, thus the characteristic function F (s) gives (a), which is described by the Hamiltonian Ĥs = e 1 |e 1 e 1 | + e 2 |e 2 e 2 | (the ground state energy is set as 0).The transition pathways |e 1 ↔ |g and |e 2 ↔ |g are coupled with two independent bosonic heat baths ( Ĥb,i = k ω i,k b † i,k bi,k , for i = h, c), and their interaction Hamiltonians are Ĥsb,i = τ + i • k g i,k bi,k +h.c., with τ + h(c) := |e 2(1) g| = [τ − h(c) t e x i t s h a 1 _ b a s e 6 4 = " i n a z e s / / y e Q k L e g 3 w L h o e W 3 G m 2 e 7 L d x r P l V l x O 9 p 8 M k s 3 B 0 7 d u Y T b J I l b I Q / K I P C Y J e U 5 2 y G u y R 0 a E k l N y R j 6 T L 8 G n 4 G v w L f i + o A a 9 8 5 7 7 x I v g x 1 8 m 2 Q x 9 < / l a t e x i t > |gi < l a t e x i t s h a 1 _ b a s e 6 4 = " g I r C u 1 4 m 5 a K D W j 6 w j s 7 A a d i 4 y T F L X O G 6 O 7 u p y P 1 2 5 l T p e w l L 8 t L N C M U i d x G P 7 p r 1 A R y 7 3 7 8 t Q d 8 G G p k 1 u u u g x m M p P p w u W k t m G 9 w z 9 c 2 j c 8 Z 1 P u J p D h t a 1 D 0 6 B 1 + 7 R t / M + P v m k d e M K h 3 H P w f B h Y u X L l 9 Z u h p e u 3 7 j 5 q 3 h 8 u 0 d V T U S k w m u W C X 3 c l C E U U E m m m p G 9 m p J g O e M 7 O a H z 3 x 8 9 4 h I R S s x 1 m 1 N M g 4 H g h Y U g 3 b Q d L i S l q B N q q G x 0 5 S D L i U 3 p X 1 j 0 p r b 6 X A 1 H s W d R e e d 5 N R Z 3 b z 7 J 3 / 3 + c v X 7 e n y 4 H c 6 q 3 d z / w 4 G Z s 3 L r d f W Y K 5 a a 7 g 1 w k v 8 b 4 E q X P 4 4 6 f K V N e 4 L w + 5 O N r w 9 W l z F e W f n w S h Z H z 1 8 5 Q 5 m H c 1 t C a 2 g e + g + S t B j t I l e o G 0 0 Q R i 9 R e / R R / Q p + B B 8 C 7 4 H P + a p w e C 0 5 g 7 q W f D r L 9 N 9 E W o = < / l a t e x i t > ø± h < l a t e x i t s h a 1 _ b a s e 6 4 = " Y t E Q B A t I z d d p 9 W G V F R H A U 1 d O 3 5 w = " > A A A D J 3 i c d V J L b 9 Q w E P a G V w m v L d y o h C K q S p x W G w S l v R X B g U t F g d 2 2 U h N W E + + k a 9 V 2 I t t p i a w c O P B b E D e 4 8 S + 4 I T h y 5 s o P w M 5 W K 9 K K i S K N v p n x 9 8 0 j K z n T Z j j 8 2 Q s u X L x 0 + c r S 1 f D a 9 R s 3 b / W X b + / q o l I U x 7 T g h d r P Q C N n E s e G G Y 7 7 p U I Q G c e 9 7 O i Z j + 8 d o 9 K s k C N T l 5 g K y 9 e 1 d Z / v t + 9 k u 4 n N 5 a u 4 m O w / G E Z b w 4 e v w s H O F l n E C r l H 7 p M N E p F H Z I e 8 I H t k T C g 5 J e / J R / L J + + B 9 9 b 5 5 3 x d U r 3 f W c 5 d 0 w v v x F y D W D U k = < / l a t e x i t > ae± < l a t e x i t s h a 1 _ b a s e 6 4 = " T a 5 b 1 E u w 6 J e z b z O S 9 n k K I 2 3 c o p 8 M a 8 3 n v b G J q r u j F 5 Y 0 R b 4 n 8 T V G b 5 w 6 j j q 8 b Y z / e 7 O 9 l p 7 e n i K i 4 7 B 5 u D 6 N n g y R t 7 M J t k b k v k A X l I N k h E t s g u e U X 2 y Y h Q c k Y + k s / k i / f J + + Z 9 9 3 7 M q V 7 v P O c + c c z 7 9 R d I 8 Q 0 k < / l a t e x i t > |g, n + 1i < l a t e x i t s h a 1 _ b a s e 6 4 = " I d q U w c + K Y r q T w T B B E P 8 S D P V q k e 4 = " > A A A D I a T s c F K 8 F 0 y 3 u J d h 0 S 9 2 z m c 1 7 O I E V t u p V T 4 I 1 5 v f e 2 M T R X d W P y x o i 2 x P 8 m q M z y h 1 H H V 4 2 x n + 9 3 d 7 L T 2 t P F V V x 2 D j Y H 0 b P B k z f 2 Y D b J 3 J b I A / K Q b J C I b J F d 8 o r s k x G h 5 I x 8 J J / J F + + T 9 8 3 7 7 v 2 Y U 7 3 e e c 5 9 4 p j 3 6 y 9 O T w 0 m < / l a t e x i t > |g, n °1i < l a t e x i t s h a 1 _ b a s e 6 4 = " K h E t 0 m T 6 y L K 9 0 r 3 / p s p 7 1 P W t G u 4 = " > A A A D I n i c d V L N b t Q wE P a G v x J + u g W p F y 4 R V a U e Y J V U Q N t b E R y 4 V B S x 2 1 Z q o t X E O 9 m 1 6 j i R 7 V A i N 8 / A O y B u 8 B Q c u S F O S D w G 3 H G y 1 Q q 3 Y q J I o 2 + + 8 T d / a c m Z 0 m H 4 s + d d u X r t + o 2 l m / 6 t 2 3 f u L v d X 7 h 2 o o p I U R 7 T g h T x K Q S F n A k e a a Y 5 H p U T I U 4 6 H 6 c m L N n 7 4 D q V i h R j q u s Q k h 6 l g G a O g L T T u r 5 7 F O e i Z y g w 2 j 8 T j K J Y g p h z H / b V w E H Y W X H a i c 2 d t d z X e + P P 1 Q 7 w / X u n 9 j i c F r X I U m n J Q 6 j g K S 5 0 Y k J p R j o 0 f V w p L o C c w x W P r C s h R J a Z r o A n W L T I J s k L a X + i g Q / / N M J A r V e e p Z X b V X o y 1 4 C K 2 7 k j p b D s x T J S V R k H n S l n F A 1 0 E 7 T i C C Z N I N a + t A 1 Q y W 2 x A Z y C B a j s 0 R 2 Z e q u 8 0 M o w S 0 1 b c P u 3 q c j t f s Z M Y X s F C / G J l R c k g M Q J P 9 f u u A U c u s x t m K F 2 w p U m V K R c 9 B VX b 6 b h g J Z h u e S / R r k P i n s 1 8 z s s Z p K h N t 3 I K v D G v 9 9 4 2 h u a q b k z e G N G W + N 8 E l V n + M O r 4 q j H 2 8 / 3 u T n Z a e 7 q 4 i s v O w e Y g e j Z 4 8 s Y e z C a Z 2 x J 5 Q B 6 S D R K R L b J L X p F 9 M i K U n J G P 5 D P 5 4 n 3 y v n n f v R 9 z q t c 7 z 7 l P H P N + / Q V I 6 Q 0 k < / l a t e x i t > |e, n °1i < l a t e x i t s h a 1 _ b a s e 6 4 = " g j f b d 2 U b u s 3 L F A 0 M 1 T + b 5 I J t P + Y = " > A A A D I n i c d V L d a t R A F J 6 N f z X + d K v Q G 2 + C p V B Q l q S o b e 8 q e u F N s e J u W 2 j C c j J 7 s j t 0 M g k z E 2 u Y 5 h l 8 B / F O n 8 J L 7 8 Q r w c f Q e y f Z s j g t n h A 4 f O c 7 8 5 2 / t O R M 6 T D 8 2 f O u X L 1 2 / c b S T f / W 7 T t 3 l / s r 9 w 5 U U U m K I 1 r w Q h 6 l o J A z g S P N N M e j s 8 e W M P Z p P M b Y k 8 I A / J B o n I F t k l r 8 g + G R F K z s h H 8 p l 8 8 T 5 5 3 7 z v 3 o 8 5 1 e u d 5 9 w n j n m / / g J D i w 0 i < / l a t e x i t > |e, n + 1i < l a t e x i t s h a 1 _ b a s e 6 4 = " c 8 5 Q A J 2 h C s I o P 2 d E F T k x s y 1 N b FIG. 1. Demonstrations for the the quantum refrigerator.(a) The transition |e1 ↔ |e2 is driven by an idealistic single mode light, which may carry nonclassical photon statistics.(b) The the whole EM field is in the canonical thermal state with temperature Te.(c) An analogue between this threelevel quantum refrigerator and the energy level structure in the sideband cooling systems, where the oscillation motion (of an ion or mechanical oscillator) is cooled down with the help of a two-level system (|e , |g ) coupled with it, and |n indicates the phonon states.
FIG. 2. (a)The optical coherence g(k) for the photon distributions(15).(b) Demonstration for the Poissonian, suband super-Poissonian statistics(15) with the same mean photon number n = 10.(c) The cooling power under different driving light obtained by Eq. (12).Here we set ξ0/γ = 1.1 and nc = 0.5, nh = 1.(d) Demonstration for the heat current blockage by a pair of bunching photons.When two photons income together, the first photon would excite the system up, but the second photon successively followed would induce the stimulated emission, which draws down the energy back before it flows to the hot bath, and that decreases the cooling power.