- Email: [email protected]
Clarifying Kirk’s confusion about quantum coherent solar cell physics via simple examples and analysis
Physica B 423 (2013) 54–57
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Clarifying Kirk’s confusion about quantum coherent solar cell physics via simple examples and analysis K.R. Chapin a,n, D. Cohen b,f, S. Das c,1, K. Dorfman d, P.K. Jha a,f, M. Kim e, A. Svidzinsky a,f, P. Vetter a,f, D.V. Voronine a a
Inst. for Quantum Science and Engineering, Texas A&M Univ., College Station, TX 77843, United States University of Toronto, Toronto, Ontario, Canada M5S 1A7 c Zur Forstquelle 6, 69126 Heidelberg, Germany d Department of Chemistry, University of California, Irvine, CA 92697, United States e Department of Physics, Louisiana State University, Baton Rouge, LA 70803, United States f College of Applied Engineering and Science, Princeton University, Princeton, NJ 08544, United States b
art ic l e i nf o
a b s t r a c t
Available online 27 March 2013
In this note, we present the ideas of quantum coherence and detailed balance in a simple and hopefully convincing way. Published by Elsevier B.V.
Keywords: Photovoltaics Solar cell Quantum coherence
1. Introduction In this note, we present the ideas of quantum coherence and detailed balance in a simple and hopefully convincing way. We will not attempt to address the overburden of misinformation in Kirk’s most recent letter [1]. However we do note that he must not be allowed to make incorrect and outrageous statements with impunity. For example, he claims to have debunked the original Physical Review Letter Ref. [2] dealing with quantum coherence and issues of detailed balance. He did nothing of the sort. He only made general (and generally incorrect) statements such as: “Scully has identified a PV cell that can generate more power than is emitted by the sun.” This nonsensical statement is simply not true. Scully made no such claim and no rational reading of Ref. [2] can come to this conclusion. Scully has correctly shown how the recombination radiation can in principle be reduced and how this can in principle enhance the open circuit voltage. Nothing is said about energy relations. Nothing is said about the conversion efficiency of flux from the sun. In fact, however, in a recent paper Dorfman et al. [3] show that the power produced by a Scully-type device in which the coherence between two conduction band levels is induced by external means, is always less than the power emitted by the sun as, of course, it must (see Fig. 1). Kirk's statement to the contrary is unsubstantiated. We will return at intervals in this note to debunk Kirk’s debunking but as stated
n
Corresponding author. Tel.: +1 979 845 1534. E-mail address: [email protected] (K.R. Chapin). 1 Currently at Max Planck Institute for Kerphysik, 69117 Heidelberg, Germany.
0921-4526/$ - see front matter Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physb.2013.03.019
earlier we prefer to concentrate on the basic physics in a way that will hopefully be more useful to the readership.
2. Photovoltaic cell Let us begin by considering the beautiful idea of Shockley and Queisser (SQ) [4], and in particular focus on a thin slice of the solar spectrum, i.e., the monochromatic cell. A particularly good place to read about this is Chapter 4 of Ref. [5]. SQ focus on the absorption and emission of light interacting with a photovoltaic (PV) cell. We consistently use the designation PV cell because high efficiency detectors are of interest to us in some ways even more than solar cells. Nevertheless, the physics of the two are quasi-identical. Consider a monochromatic slice of the solar spectrum incident on a photocell as in Fig. 2. Absorption and emission of monochromatic radiation are governed by the average photon number n s . Equation of motion for n s is given by ∂n s ∂n s j þ j n_ s ¼ ∂t abs ∂t emit 2 g ½−n s N v þ ðn s þ 1ÞN c ¼ γ Nc ; ¼ κ −n s þ N v −Nc
ð1Þ
where the coupling strength between the radiation and the electrons is given by g, the time between disruptive collisions is given by γ 1 and κ ¼ ðg 2 =γÞðN v −N c Þ. Recognizing that the population of the valence Nv and conduction Nc band levels can be described (in the grand canonical
K.R. Chapin et al. / Physica B 423 (2013) 54–57
55
This is the detailed balance analysis for the quantum efficiency (open circuit voltage). Can we “improve” the result, at least in principle? The answer is yes, by using radiatively induced quantum coherence as in Ref. [2] to reduce the rate of radiative recombination. This can be understood by studying Fig. 4. To calculate the absorption from v-c1 and v-c2 we must add probabilities. But to calculate emission from c1 -v and c2 -v (same final state), we must add probability amplitudes and then square. For such scheme the equation of motion for the average photon number n s contains additional terms appearing due to quantum interference in the emission channel g2 n_ s ¼ −2 Av j2 n s γ Fig. 1. Power of a photocell as a function of voltage with coherence induced by external microwave field Ω ¼ 50γ and different decoherence rates γ τ ¼ 0 and γ τ ¼ 50γ~ 1 . The lower curve shows photocell power with no microwave drive (Ω ¼ 0), while the upper curve indicates power acquired from the Sun. For details of the model cell parameters see Ref. [3]. For any voltage the PV cell generates less power than that acquired from the Sun.
Absorption
þ
g 2 2 2 ð A1 j þ A2 j þ A1 An2 þ A2 An1 Þðn s þ 1Þ γ
Thus if, for example, we have maximum coherence and A1 ¼ −A2 then recombination is canceled. In general the coherence is smaller than this and we write jAv j2 ¼ ρvv , jA1 j2 ¼ ρ11 , jA2 j2 ¼ ρ22 , A1 An2 ¼ ρ12 and A2 An1 ¼ ρ21 which yields 1 n_ s ¼ κ~ n s − x e −1
c s
v
c
Ta eV ¼ ℏν 1− þ δ: Tc
v Fig. 2. Photon absorption and emission processes between valence and conduction bands and their rates. Here n s is the average number of photons in the thin slice of solar spectrum.
Nc T
v,
Nv
ð6Þ
where ex ¼ 2ρvv =½ρ11 þ ρ22 þ ρ12 þ ρ21 . In most systems, ρ12 -0 due to environmental decoherence. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi However, if the effective coherent drive frequency is g d nd þ 1 where ℏg d is the electrondriven field coupling energy and nd is the number of photons in the drive mode, then this allows us to have a (small) increase δ of the open circuit voltage
Emission
c,
ð5Þ
µc µv
Fig. 3. Nv(Nc) electrons in the valence (conduction) band of energy ϵc ðϵc Þ decrease the number of photons by absorption (increase the number of photons by radiative recombination).
ð7Þ
This increase comes at a cost as in the case of the photo-Carnot quantum heat engine of Fig. 5. Here the photon working fluid may actually yield more energy using a phase coherent ensemble of atoms (fuel), and the efficiency may exceed the Carnot efficiency [6]. This is a good example of how quantum thermodynamics can go beyond classical thermodynamics. It is also a common example of Kirk's confusion. He says “The new working fluid [in the photoCarnot engine] is phaseonium” [7]. As is noted above, the photons are the working fluid (like steam) and the fuel is phaseonium (like coal in a steam engine).
ensemble) by their chemical potentials and ambient temperature Ta (Fig. 3) we write Nv ϵc −ϵv −ðμc −μv Þ : ð2Þ ¼ exp kT a Nc The photocell voltage is eV ¼ μc −μv . If we take n s to be equal to 1=ðeℏν=kT s −1Þ, where TS is the solar temperature, then Eqs. (1) and (2) yield 1 1 : ð3Þ n_ s ¼ κ~ ℏν=kT − ðℏν−eVÞ=kT s a −1 e −1 e For the open circuit (no current in the system) Eq. (3) gives the open circuit voltage Ta eV ¼ ℏν 1− ð4Þ Ts obtained by setting n_ s ¼ 0 and solving for V.
Fig. 4. Photon absorption and emission in a model with coherence between upper levels c1 and c2. Av, A1 and A2 are probability amplitudes to find the system in states v, c1 and c2, respectively, and g1 (g2) is the coupling constant between state v and state 1 (2). Quantum interference in the emission channel arises because the emission rate goes as jg1 A1 þ g2 A2 j2 . There is no interference in the absorption channel since the absorption rate is proportional to jg1 Av j2 þ jg2 Av j2 .
56
K.R. Chapin et al. / Physica B 423 (2013) 54–57
laser frequency νℓ
Tc ℏνℓ ¼ ℏνh −ℏνc ¼ ℏνh 1− : Th
ð11Þ
This allows us to increase ℏνℓ and even achieve lasing without population inversion which as been a topic of research interest during last two decades [9–11].
4. Photo-Carnot quantum heat engine
Fig. 5. (A) Photo-Carnot engine in which radiation pressure from a thermally excited single-mode field drives a piston. Atoms flow through the engine and keep the field at a constant temperature Trad for the isothermal 1-2 portion of the Carnot cycle. Upon exiting the engine, the bath atoms are cooler than when they entered and are reheated by interactions with the hohlraum at Th and “stored” in the preparation for the next cycle. The combination of reheating and storing is depicted in (A) as the heat reservoir. A cold reservoir at Tc provides the entropy sink. (B) Two-level atoms in a regular thermal distribution, determined by temperature Th, heat the driving radiation to T rad ¼ T h such that the regular operating efficiency is given by η. (C) When the field is heated, however, by a phaseonium in which the ground state doublet has a small amount of coherence and the populations of levels a, b, and c, are thermally distributed, the field temperature is T rad 4 T h , and the operating efficiency is given by ηϕ ¼ η−π cos Φ.
Fig. 6. Thermal radiation from a “hot” source (temperature Th, frequency νh ) constitutes the energy reservoir while “cold” (temperature Tc, frequency νc ) forms the entropy sink. The combination produces coherent laser radiation at the a−b transition, i.e., useful work.
3. Laser as a quantum heat engine The laser is a heat engine with Carnot quantum efficiency [8]. To demonstrate this we consider a model in which an incoherent thermal “hot” radiation at temperature Th and energy ℏνh serves as the energy source that populates the upper laser level a (see Fig. 6). The lower laser level b is coupled to the ground state c by “cold” light at Tc and energy ℏνc which serves as an entropy sink. The average number of hot and cold thermal photons is given by the Planck distribution nh ¼
1 ; expðℏνh =kT h Þ−1
nc ¼
1 : expðℏνc =kT c Þ−1
ð8Þ
The rate equation for the number of atoms Na in the upper state a reads N_ a ¼ −ðnh þ 1ÞN a þ nh N c ; κh
ð9Þ
where Nc is the number of atoms in the ground state c. Taking N_ a to be zero Eq. (9) yields Na =Nc ¼ nh =ðnh þ 1Þ ¼ e−ℏνh =kT h . The rate equation for the number of atoms in the lower state b is given by N_ b ¼ −ðnc þ 1ÞNb þ nc N c κc
In [6] a quantum Carnot engine was developed in which the atoms in the heat bath are given a small bit of quantum coherence. The induced quantum coherence becomes vanishingly small in the high-temperature limit and the heat bath is essentially thermal. However, the phase ϕ, associated with the atomic coherence, provides a new control parameter. The deep physics behind the second law of thermodynamics was not violated; however, the quantum Carnot engine did have certain features that were not possible in a classical engine. Specifically in Ref. [6] a new kind of quantum Carnot engine powered by a special quantum heat bath consisting of phase coherent atoms (a.k.a. phaseonium) was proposed and analyzed which allowed for the extraction of work from a single thermal reservoir. In this specific heat engine, it was the radiation pressure which drove the piston. Thus the radiation was the working fluid which was heated by a beam of hot atoms (Fig. 5). Thus, one can get work from a single bath by using quantum coherence. This is possible because quantum coherence allows for the breaking of detailed balance between emission and absorption as in the case of lasing without inversion. It is not a “perpetual mobile of the second kind”. It does extract work from a single heat bath; but it takes energy, e.g., from an external source of microwaves, to prepare the coherence. Alternatively, a microwave generator (which produces the coherence) could be incorporated into the photo-Carnot engine sort of like a “quantum supercharger.” The practicality and utility of this photo-Carnot engine are not the issue. But rather it is the fact that quantum control via quantum coherence provides a new tool in the study of thermodynamics. In short, quantum coherence provides a control parameter that allows work to be extracted from a single heat bath. The total system entropy is constantly increasing, and the physics behind the second law is not violated. However, quantum coherence does allow certain features of quantum engine operation not possible with a classical heat engine. In summary, the fallacy and fundamental erroneousness of Kirk’s claim that quantum coherence cannot increase photocell power have been clearly explained, for the fourth time now [12–14]. In the last few years we have shown that quantum coherence can result in new inroads into photocell engineering by increasing cell power [2,3,12-14] and yielding new insights in photosynthesis [15]. We hope that present discussion will help to bridge the gap between the quantum optics, quantum thermodynamics, and solar cell physics communities.
ð10Þ
which in steady state gives Nb =Nc ¼ nc =ðnc þ 1Þ ¼ e−ℏνc =kT c . At lasing threshold we have 1 ¼ N a =N b ¼ Na =Nc N c =N b which yields ℏνh =kT h ¼ ℏνc =kT c . As a result, we obtain Carnot formula for the
Acknowledgments We acknowledge the support of the National Science Foundation PHY-1241032 and the Robert A. Welch Foundation (Awards A-1261). References [1] A.P. Kirk, Physica B 407 (2012) 544. [2] M.O. Scully, Phys. Rev. Lett. 104 (2010) 207701. [3] K. Dorfman, M. Kim, A.A. Svidzinsky, Phys. Rev. A 84 (2011) 053829.
K.R. Chapin et al. / Physica B 423 (2013) 54–57
[4] W. Shockley, H.J. Queisser, J. Appl. Phys. 32 (1961) 510. [5] Antonio Luque, Antonio Martí, Theoretical limits of photovoltaic conversion and new-generation solar cells, in: Antonio Luque, Steven Hegedus (Eds.), Handbook of Photovoltaic Science and Engineering, John Wiley & Sons Ltd, West Sussex, UK, 2011. (Chapter 4). [6] M.O. Scully, S. Zubairy, G. Agarwal, H. Walther, Science 299 (2003) 862. [7] A.P. Kirk, Phys. Rev. Lett. 106 (2011) 048703. [8] D. Scovil, E. Schulz-DuBois, Phys. Rev. Lett. 2 (1959) 262. [9] O. Kocharovskaya, Ya.I. Khanin, Zh. Eksp. Teor. Phys. 90 (1986) 1610; O. Kocharovskaya, Ya.I. Khanin, JETP 63 (1986) 945.
57
[10] S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033. [11] M.O. Scully, M.S. Zubairy, Quantum Optics Cambridge, University Press Cambridge, UK, 1997. [12] M.O. Scully, Phys. Rev. Lett. 106 (2011) 049801. [13] M.O. Scully, K.R. Chapin, K.E. Dorfman, M.B. Kim, A. Svidzinsky, Proc. Natl. Acad. Sci. 108 (2011) 15097. [14] K.R. Chapin, D. Cohen, S. Das, K. Dorfman, P.K. Jha, M. Kim, A. Svidzinsky, P. Vetter, D.V. Voronine, Physica B 417 (2013) 91. [15] K.E. Dorfman, D.V. Voronine, S. Mukamel, M.O. Scully, Proc. Natl. Acad. Sci. 110 (2013) 2746.
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Clarifying Kirk’s confusion about quantum coherent solar cell physics via simple examples and analysis K.R. Chapin a,n, D. Cohen b,f, S. Das c,1, K. Dorfman d, P.K. Jha a,f, M. Kim e, A. Svidzinsky a,f, P. Vetter a,f, D.V. Voronine a a
Inst. for Quantum Science and Engineering, Texas A&M Univ., College Station, TX 77843, United States University of Toronto, Toronto, Ontario, Canada M5S 1A7 c Zur Forstquelle 6, 69126 Heidelberg, Germany d Department of Chemistry, University of California, Irvine, CA 92697, United States e Department of Physics, Louisiana State University, Baton Rouge, LA 70803, United States f College of Applied Engineering and Science, Princeton University, Princeton, NJ 08544, United States b
art ic l e i nf o
a b s t r a c t
Available online 27 March 2013
In this note, we present the ideas of quantum coherence and detailed balance in a simple and hopefully convincing way. Published by Elsevier B.V.
Keywords: Photovoltaics Solar cell Quantum coherence
1. Introduction In this note, we present the ideas of quantum coherence and detailed balance in a simple and hopefully convincing way. We will not attempt to address the overburden of misinformation in Kirk’s most recent letter [1]. However we do note that he must not be allowed to make incorrect and outrageous statements with impunity. For example, he claims to have debunked the original Physical Review Letter Ref. [2] dealing with quantum coherence and issues of detailed balance. He did nothing of the sort. He only made general (and generally incorrect) statements such as: “Scully has identified a PV cell that can generate more power than is emitted by the sun.” This nonsensical statement is simply not true. Scully made no such claim and no rational reading of Ref. [2] can come to this conclusion. Scully has correctly shown how the recombination radiation can in principle be reduced and how this can in principle enhance the open circuit voltage. Nothing is said about energy relations. Nothing is said about the conversion efficiency of flux from the sun. In fact, however, in a recent paper Dorfman et al. [3] show that the power produced by a Scully-type device in which the coherence between two conduction band levels is induced by external means, is always less than the power emitted by the sun as, of course, it must (see Fig. 1). Kirk's statement to the contrary is unsubstantiated. We will return at intervals in this note to debunk Kirk’s debunking but as stated
n
Corresponding author. Tel.: +1 979 845 1534. E-mail address: [email protected] (K.R. Chapin). 1 Currently at Max Planck Institute for Kerphysik, 69117 Heidelberg, Germany.
0921-4526/$ - see front matter Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physb.2013.03.019
earlier we prefer to concentrate on the basic physics in a way that will hopefully be more useful to the readership.
2. Photovoltaic cell Let us begin by considering the beautiful idea of Shockley and Queisser (SQ) [4], and in particular focus on a thin slice of the solar spectrum, i.e., the monochromatic cell. A particularly good place to read about this is Chapter 4 of Ref. [5]. SQ focus on the absorption and emission of light interacting with a photovoltaic (PV) cell. We consistently use the designation PV cell because high efficiency detectors are of interest to us in some ways even more than solar cells. Nevertheless, the physics of the two are quasi-identical. Consider a monochromatic slice of the solar spectrum incident on a photocell as in Fig. 2. Absorption and emission of monochromatic radiation are governed by the average photon number n s . Equation of motion for n s is given by ∂n s ∂n s j þ j n_ s ¼ ∂t abs ∂t emit 2 g ½−n s N v þ ðn s þ 1ÞN c ¼ γ Nc ; ¼ κ −n s þ N v −Nc
ð1Þ
where the coupling strength between the radiation and the electrons is given by g, the time between disruptive collisions is given by γ 1 and κ ¼ ðg 2 =γÞðN v −N c Þ. Recognizing that the population of the valence Nv and conduction Nc band levels can be described (in the grand canonical
K.R. Chapin et al. / Physica B 423 (2013) 54–57
55
This is the detailed balance analysis for the quantum efficiency (open circuit voltage). Can we “improve” the result, at least in principle? The answer is yes, by using radiatively induced quantum coherence as in Ref. [2] to reduce the rate of radiative recombination. This can be understood by studying Fig. 4. To calculate the absorption from v-c1 and v-c2 we must add probabilities. But to calculate emission from c1 -v and c2 -v (same final state), we must add probability amplitudes and then square. For such scheme the equation of motion for the average photon number n s contains additional terms appearing due to quantum interference in the emission channel g2 n_ s ¼ −2 Av j2 n s γ Fig. 1. Power of a photocell as a function of voltage with coherence induced by external microwave field Ω ¼ 50γ and different decoherence rates γ τ ¼ 0 and γ τ ¼ 50γ~ 1 . The lower curve shows photocell power with no microwave drive (Ω ¼ 0), while the upper curve indicates power acquired from the Sun. For details of the model cell parameters see Ref. [3]. For any voltage the PV cell generates less power than that acquired from the Sun.
Absorption
þ
g 2 2 2 ð A1 j þ A2 j þ A1 An2 þ A2 An1 Þðn s þ 1Þ γ
Thus if, for example, we have maximum coherence and A1 ¼ −A2 then recombination is canceled. In general the coherence is smaller than this and we write jAv j2 ¼ ρvv , jA1 j2 ¼ ρ11 , jA2 j2 ¼ ρ22 , A1 An2 ¼ ρ12 and A2 An1 ¼ ρ21 which yields 1 n_ s ¼ κ~ n s − x e −1
c s
v
c
Ta eV ¼ ℏν 1− þ δ: Tc
v Fig. 2. Photon absorption and emission processes between valence and conduction bands and their rates. Here n s is the average number of photons in the thin slice of solar spectrum.
Nc T
v,
Nv
ð6Þ
where ex ¼ 2ρvv =½ρ11 þ ρ22 þ ρ12 þ ρ21 . In most systems, ρ12 -0 due to environmental decoherence. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi However, if the effective coherent drive frequency is g d nd þ 1 where ℏg d is the electrondriven field coupling energy and nd is the number of photons in the drive mode, then this allows us to have a (small) increase δ of the open circuit voltage
Emission
c,
ð5Þ
µc µv
Fig. 3. Nv(Nc) electrons in the valence (conduction) band of energy ϵc ðϵc Þ decrease the number of photons by absorption (increase the number of photons by radiative recombination).
ð7Þ
This increase comes at a cost as in the case of the photo-Carnot quantum heat engine of Fig. 5. Here the photon working fluid may actually yield more energy using a phase coherent ensemble of atoms (fuel), and the efficiency may exceed the Carnot efficiency [6]. This is a good example of how quantum thermodynamics can go beyond classical thermodynamics. It is also a common example of Kirk's confusion. He says “The new working fluid [in the photoCarnot engine] is phaseonium” [7]. As is noted above, the photons are the working fluid (like steam) and the fuel is phaseonium (like coal in a steam engine).
ensemble) by their chemical potentials and ambient temperature Ta (Fig. 3) we write Nv ϵc −ϵv −ðμc −μv Þ : ð2Þ ¼ exp kT a Nc The photocell voltage is eV ¼ μc −μv . If we take n s to be equal to 1=ðeℏν=kT s −1Þ, where TS is the solar temperature, then Eqs. (1) and (2) yield 1 1 : ð3Þ n_ s ¼ κ~ ℏν=kT − ðℏν−eVÞ=kT s a −1 e −1 e For the open circuit (no current in the system) Eq. (3) gives the open circuit voltage Ta eV ¼ ℏν 1− ð4Þ Ts obtained by setting n_ s ¼ 0 and solving for V.
Fig. 4. Photon absorption and emission in a model with coherence between upper levels c1 and c2. Av, A1 and A2 are probability amplitudes to find the system in states v, c1 and c2, respectively, and g1 (g2) is the coupling constant between state v and state 1 (2). Quantum interference in the emission channel arises because the emission rate goes as jg1 A1 þ g2 A2 j2 . There is no interference in the absorption channel since the absorption rate is proportional to jg1 Av j2 þ jg2 Av j2 .
56
K.R. Chapin et al. / Physica B 423 (2013) 54–57
laser frequency νℓ
Tc ℏνℓ ¼ ℏνh −ℏνc ¼ ℏνh 1− : Th
ð11Þ
This allows us to increase ℏνℓ and even achieve lasing without population inversion which as been a topic of research interest during last two decades [9–11].
4. Photo-Carnot quantum heat engine
Fig. 5. (A) Photo-Carnot engine in which radiation pressure from a thermally excited single-mode field drives a piston. Atoms flow through the engine and keep the field at a constant temperature Trad for the isothermal 1-2 portion of the Carnot cycle. Upon exiting the engine, the bath atoms are cooler than when they entered and are reheated by interactions with the hohlraum at Th and “stored” in the preparation for the next cycle. The combination of reheating and storing is depicted in (A) as the heat reservoir. A cold reservoir at Tc provides the entropy sink. (B) Two-level atoms in a regular thermal distribution, determined by temperature Th, heat the driving radiation to T rad ¼ T h such that the regular operating efficiency is given by η. (C) When the field is heated, however, by a phaseonium in which the ground state doublet has a small amount of coherence and the populations of levels a, b, and c, are thermally distributed, the field temperature is T rad 4 T h , and the operating efficiency is given by ηϕ ¼ η−π cos Φ.
Fig. 6. Thermal radiation from a “hot” source (temperature Th, frequency νh ) constitutes the energy reservoir while “cold” (temperature Tc, frequency νc ) forms the entropy sink. The combination produces coherent laser radiation at the a−b transition, i.e., useful work.
3. Laser as a quantum heat engine The laser is a heat engine with Carnot quantum efficiency [8]. To demonstrate this we consider a model in which an incoherent thermal “hot” radiation at temperature Th and energy ℏνh serves as the energy source that populates the upper laser level a (see Fig. 6). The lower laser level b is coupled to the ground state c by “cold” light at Tc and energy ℏνc which serves as an entropy sink. The average number of hot and cold thermal photons is given by the Planck distribution nh ¼
1 ; expðℏνh =kT h Þ−1
nc ¼
1 : expðℏνc =kT c Þ−1
ð8Þ
The rate equation for the number of atoms Na in the upper state a reads N_ a ¼ −ðnh þ 1ÞN a þ nh N c ; κh
ð9Þ
where Nc is the number of atoms in the ground state c. Taking N_ a to be zero Eq. (9) yields Na =Nc ¼ nh =ðnh þ 1Þ ¼ e−ℏνh =kT h . The rate equation for the number of atoms in the lower state b is given by N_ b ¼ −ðnc þ 1ÞNb þ nc N c κc
In [6] a quantum Carnot engine was developed in which the atoms in the heat bath are given a small bit of quantum coherence. The induced quantum coherence becomes vanishingly small in the high-temperature limit and the heat bath is essentially thermal. However, the phase ϕ, associated with the atomic coherence, provides a new control parameter. The deep physics behind the second law of thermodynamics was not violated; however, the quantum Carnot engine did have certain features that were not possible in a classical engine. Specifically in Ref. [6] a new kind of quantum Carnot engine powered by a special quantum heat bath consisting of phase coherent atoms (a.k.a. phaseonium) was proposed and analyzed which allowed for the extraction of work from a single thermal reservoir. In this specific heat engine, it was the radiation pressure which drove the piston. Thus the radiation was the working fluid which was heated by a beam of hot atoms (Fig. 5). Thus, one can get work from a single bath by using quantum coherence. This is possible because quantum coherence allows for the breaking of detailed balance between emission and absorption as in the case of lasing without inversion. It is not a “perpetual mobile of the second kind”. It does extract work from a single heat bath; but it takes energy, e.g., from an external source of microwaves, to prepare the coherence. Alternatively, a microwave generator (which produces the coherence) could be incorporated into the photo-Carnot engine sort of like a “quantum supercharger.” The practicality and utility of this photo-Carnot engine are not the issue. But rather it is the fact that quantum control via quantum coherence provides a new tool in the study of thermodynamics. In short, quantum coherence provides a control parameter that allows work to be extracted from a single heat bath. The total system entropy is constantly increasing, and the physics behind the second law is not violated. However, quantum coherence does allow certain features of quantum engine operation not possible with a classical heat engine. In summary, the fallacy and fundamental erroneousness of Kirk’s claim that quantum coherence cannot increase photocell power have been clearly explained, for the fourth time now [12–14]. In the last few years we have shown that quantum coherence can result in new inroads into photocell engineering by increasing cell power [2,3,12-14] and yielding new insights in photosynthesis [15]. We hope that present discussion will help to bridge the gap between the quantum optics, quantum thermodynamics, and solar cell physics communities.
ð10Þ
which in steady state gives Nb =Nc ¼ nc =ðnc þ 1Þ ¼ e−ℏνc =kT c . At lasing threshold we have 1 ¼ N a =N b ¼ Na =Nc N c =N b which yields ℏνh =kT h ¼ ℏνc =kT c . As a result, we obtain Carnot formula for the
Acknowledgments We acknowledge the support of the National Science Foundation PHY-1241032 and the Robert A. Welch Foundation (Awards A-1261). References [1] A.P. Kirk, Physica B 407 (2012) 544. [2] M.O. Scully, Phys. Rev. Lett. 104 (2010) 207701. [3] K. Dorfman, M. Kim, A.A. Svidzinsky, Phys. Rev. A 84 (2011) 053829.
K.R. Chapin et al. / Physica B 423 (2013) 54–57
[4] W. Shockley, H.J. Queisser, J. Appl. Phys. 32 (1961) 510. [5] Antonio Luque, Antonio Martí, Theoretical limits of photovoltaic conversion and new-generation solar cells, in: Antonio Luque, Steven Hegedus (Eds.), Handbook of Photovoltaic Science and Engineering, John Wiley & Sons Ltd, West Sussex, UK, 2011. (Chapter 4). [6] M.O. Scully, S. Zubairy, G. Agarwal, H. Walther, Science 299 (2003) 862. [7] A.P. Kirk, Phys. Rev. Lett. 106 (2011) 048703. [8] D. Scovil, E. Schulz-DuBois, Phys. Rev. Lett. 2 (1959) 262. [9] O. Kocharovskaya, Ya.I. Khanin, Zh. Eksp. Teor. Phys. 90 (1986) 1610; O. Kocharovskaya, Ya.I. Khanin, JETP 63 (1986) 945.
57
[10] S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033. [11] M.O. Scully, M.S. Zubairy, Quantum Optics Cambridge, University Press Cambridge, UK, 1997. [12] M.O. Scully, Phys. Rev. Lett. 106 (2011) 049801. [13] M.O. Scully, K.R. Chapin, K.E. Dorfman, M.B. Kim, A. Svidzinsky, Proc. Natl. Acad. Sci. 108 (2011) 15097. [14] K.R. Chapin, D. Cohen, S. Das, K. Dorfman, P.K. Jha, M. Kim, A. Svidzinsky, P. Vetter, D.V. Voronine, Physica B 417 (2013) 91. [15] K.E. Dorfman, D.V. Voronine, S. Mukamel, M.O. Scully, Proc. Natl. Acad. Sci. 110 (2013) 2746.