Work distributions on quantum fields

Work distributions on Quantum Fields Author: Álvaro Ortega González Advisor: Eduardo Martín-Martínez UPC tutor: Alejandro Pozas Kerstjens Grado en Matemáticas. Grado en Ingeniería Física. May 2019 Abstract In this thesis we tackle the problem of defining a notion of work distribution for nonequilibrium processes occurring in quantum fields. We study the work cost of processes in quantum fields without the need of projective measurements, which are always ill-defined in quantum field theory. Inspired by interferometry schemes, we propose a work distribution that generalizes the two-point measurement scheme employed in quantum thermodynamics to the case of quantum fields and avoids the use of projective measurements. The distribution is calculated for local unitary processes performed on KMS (thermal) states of scalar fields. Crooks theorem and the Jarzynski equality are shown to be satisfied, and some features of the resulting distributions are studied as functions of temperature and the degree of spatiotemporal localization of the unitary operation. We show how the work fluctuations become much larger than the average as the process becomes more localized in both time and space. This thesis led to one publication [1] The thesis is structured as follows: In Chapter 1, a review of perturbation theory and some aspects of quantum field theory are included for completeness. In Chapters 2 and 3, we introduce the concepts of quantum thermodynamics and relativistic quantum information that we use in the derivations of our results. Chapter 4 is a summary of [1], which is included in the Appendix. Keywords: Quantum fields, thermodynamics, causality, interferometry Publications forming part of the thesis •Work distributions on quantum fields,´ Alvaro Ortega, Emma McKay, ´ Alvaro M. Alhambra, Eduardo Mart´ın-Mart´ınez, arXiv:1902.03258 1 Acknowledgments I want to thank Eduardo Mart´ın-Mart´ınez for letting me join his group for my Bachelor Thesis, and for giving me such a beautiful project to work on. It was a pleasure for me to be part of Barrio for the past 8 months. I am also grateful to Emma McKay and ´ Alvaro M. Alhambra for all the things that I have learned from them. They have been tremendous people to look up to. Finally, I would also like to thank CFIS and Fundaci´o Privada Cellex, for making this Mobility Program possible. 2 Contents 1 Preliminaries 4 1.1 PerturbationTheory................................ 4 1.2 QuantumFieldTheory............................... 6 2 Quantum Thermodynamics 9 2.1 Classical Non-equilibrium Statistical Physics . . . . . . . . . . . . . . . . . . 9 2.2 Quantum non-equilibrium statistical physics . . . . . . . . . . . . . . . . . . . 11 2.3 KMSstates ..................................... 14 3 Relativistic Quantum Information 17 3.1 Measuring quantum fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Work distributions on quantum fields 21 Bibliography 23 Appendix: Work distributions on quantum fields (Article) 26 3 Chapter 1 Preliminaries Throughout all this thesis, we work in units in which c=~=kB= 1 In this chapter we introduce the fundamental notions of perturbation theory and quantum field theory that are used throughout our work. There are many good books [2, 3] which explain in more detail and more thoroughly these concepts, that we review here for completeness. 1.1 Perturbation Theory At the heart of Quantum Mechanics is the Schrodinger equation: id dt |ψ(t)i=ˆ H(t)|ψ(t)i(1.1) which describes the time evolution of quantum mechanical systems. It can easily be proven that the evolution generated by (1.1) is unitary: |ψ(t)i=ˆ U(t, t0)|ψ(t0)i,(1.2) where ˆ U(t, t0) is a unitary operator satisfying ˆ U(t0, t0) = Iand |ψ(t0)iis some initial condition. Using (1.1) a differential equation for the unitary operator ˆ U(t, t0) can be obtained as well id dt ˆ U(t, t0) = ˆ H(t)ˆ U(t, t0) (1.3) Despite being widely studied, the Schrodinger equation is yet far from being fully understood. Analytical solutions are only known for a handful of problems which describe very simple systems, involving few non-interacting particles and described by time-independent Hamiltonians ˆ H(t) = ˆ H0. There are many problems of interest in which the Hamiltonian of the system can be written as a constant term plus a small perturbation, ˆ H(t) = ˆ H0+λˆ V(t), where λis a small constant. In these cases, approximate solutions with different levels of accuracy can be obtained by expressing the unitary operator that dictates the time evolution as a power series in λ. We review here the theory behind this procedure, since in our work we will consider evolutions of quantum fields generated by perturbed Hamiltonians. The Schrodinger and Heisenberg pictures In an introductory Quantum Mechanics course, two ways of interpreting the time evolution of a quantum system are usually presented: the Schrodinger and the Heisenberg pictures. In the Schrodinger picture, the state vectors evolve with time while the operators are time independent: |ψ(t)i=ˆ U(t, t0)|ψ(t0)i.(1.4) The equation of motion obeyed by the state vectors is (1.1). In the Heisenberg picture, the operators are the objects that change with time, with the state vectors remaining time independent. More precisely, if ˆ U(t, t0) is the unitary evolution generated by the Hamiltonan ˆ H(t) between times t0and t, according to equation (1.1), and ˆ O(t) is a given operator, we would have that ˆ OH(t) = ˆ U†(t, t0)ˆ O(t)ˆ U(t, t0).(1.5) 4 ˆ OH(t) is the Heisenberg operator corresponding to the operator ˆ O(t), which corresponds to the operator in the Schrodinger picture. Note that we have included the possibility of ˆ O(t) being explicitly time dependent. We now obtain the equation of motion for ˆ OH(t). Differentiating (1.5) and using (1.3) we obtain d dt ˆ OH(t) = i(ˆ U†ˆ Hˆ Oˆ U−ˆ U†ˆ Oˆ Hˆ U) + ˆ U†∂ˆ O ∂t ˆ U. (1.6) Rewriting the right hand side in terms of the operators ˆ Oand ˆ Hwritten in the Heisenberg picture yields d dt ˆ OH(t) = ihˆ HH(t),ˆ OH(t)i+∂ˆ O ∂t H.(1.7) The time derivative of the expectation value of an observable ˆ Ohas a similar structure in both the Schrodinger and Heisenberg pictures. We start by computing it in the Schrodinger picture: d dt Dˆ OEt=∂ ∂t hψ(t)|ˆ O|ψ(t)i+hψ(t)|ˆ O∂ ∂t |ψ(t)i+hψ(t)|∂ˆ O ∂t |ψ(t)i.(1.8) Using (1.1) gives d dt Dˆ OEt=ihψ(t)|(ˆ Hˆ O−ˆ Oˆ H|ψ(t)i+hψ(t)|∂ˆ O ∂t |ψ(t)i.(1.9) Therefore, d dt Dˆ OEt=iDhˆ H, ˆ OiEt+*∂ˆ O ∂t +t .(1.10) To obtain the equivalent expression in the Heisenberg picture, the procedure is analogous. In this case we will use (1.3) instead of (1.1). d dt Dˆ OEt=hψ(t0)|dˆ OH dt |ψ(t0)i=iDhˆ HH,ˆ OHiEt0 +*∂ˆ O ∂t H+t0 .(1.11) The Dirac Picture We are now going to introduce a framework that will be very useful to treat time-dependent problems. It is in between the Schrodinger and Heisenberg pictures, and it is known as the interaction or the Dirac picture. Both the states and the operators will depend on time, but on a carefully chosen way. Let the Hamiltonian of the system be ˆ H(t) = ˆ H0+λˆ V(t),(1.12) and let ˆ U0=eit ˆ H0be the unitary transformation that transforms operators and states from the Schrodinger picture to the Dirac picture in the following way: ˆ OD(t) = ˆ U0ˆ OS(t)ˆ U† 0,(1.13) |ψ(t)iD=ˆ U0|ψs(t)i,(1.14) where the subscripts Dand Sare used to denote operators and states in the Dirac and Schrodinger pictures, respectively. We will omit the subscript Sfrom now on, as it should be clear from the context when we are referring to objects in the Schrodinger picture. We now derive the expression for the Schrodinger equation in the Dirac picture. Starting from (1.1) and introducing the identity as ˆ U† 0ˆ U0we obtain, id dtˆ U† 0ˆ U0|ψ(t)i=ˆ U† 0ˆ U0ˆ H(t)ˆ U† 0ˆ U0|ψ(t)i. Using the expressions for the operators and states in the Dirac picture, (1.13) and (1.14) gives id dtˆ U† 0|ψ(t)iD=ˆ U† 0ˆ HD(t)|ψ(t)iD. 5 From here, a few steps of calculations lead to the final expression id dt |ψ(t)iD=λVD(t)|ψ(t)i.(1.15) As we can see, the time evolution for the kets in the interaction picture is generated only by the time dependent part of the Hamiltonian. Equation (1.15) has the same mathematical form as (1.1), so it should not be surprising that we can also write for a state |ψ(t)iDin the interaction picture |ψ(t)iD=ˆ UD(t, t0)|ψ(t0)iD,(1.16) where ˆ UD(t, t0) is a unitary operator. Substituting this into (1.15) we obtain a differential equation for the unitary operator ˆ UD(t, t0) d dt ˆ UD(t, t0) = −iλ ˆ VD(t)ˆ UD(t, t0),(1.17) . with the initial condition ˆ UD(t, t0) = I.(1.18) Expressing (1.15) in integral form yields ˆ UD(t, t0) = I−iλ Zt t0 dt1ˆ VD(t1)ˆ UD(t1, t0).(1.19) By recursively substituting ˆ UD(t, t0) in the right hand side of the previous equation we obtain ˆ UD(t, t0) = I−iλ Zt t0 dt1ˆ VD(t1)1−iλ Zt1 t0 dt2ˆ VD(t2)1−... (1.20) When λis much smaller than 1, from the previous expression we can derive the following power expansion for ˆ UD(t, t0) ˆ UD(t, t0) = 1 + ˆ U(1)(t, t0) + ˆ U(2)(t, t0) + ... (1.21) where ˆ U(1) D(t, t0) = −iλ Zt1 t0 dt1ˆ VD(t1),(1.22) ˆ U(2) D(t, t0) = −λ2Zt t0 dt1Zt1 t0 dt2ˆ VD(t1)ˆ VD(t2).(1.23) We will use these expression repeatedly in our work, as we will be mostly concerned with processes occurring in quantum fields arising from very small, local interactions. Usually (1.21) is written more succinctly as ˆ UD(t, t0) = Texp −iλ Zt t0 dtˆ VD(t)!,(1.24) where Trefers to the time ordering operator. 1.2 Quantum Field Theory We will now introduce the concepts of quantum field theory that are used in our work. We will first review some concepts of classical field theory and then we will show how to obtain a quantum field theory from a classical field theory by applying canonical quantization. We will mainly focus on the real scalar field, as this is the quantum field we study in our work. 6 Classical field theory We can think of a classical field as a system described by a continuum of degrees of freedom φ(t, x), where in this case the position in space xis being used to index the different degrees of freedom. Our first goal will be to derive the equations of motion that will allow us to obtain the configuration of the field φ(t, x) at a time t, given its configuration at a time t0,φ(t0,x). The dynamics of a field are governed by a Lagrangian, which is a function of φ(t, x), ˙ φ(t, x) and ∇φ(t, x). In order to preserve locality, the Lagrangian at a time twill be the integral over space of a Lagrangian density, L(t) = Rd3xL(φ, ∂µφ). Furthermore, the Lagrangian density should be covariant, if we want our equations of motion to be consistent with special relativity. The equations of motion will be derived from the principle of least action, δS = 0 (where, S=Rt2 t1dtL(t) = Rt2 t1dtRd3xL=Rd4xL). For the particular case in which the Lagrangian density is that of the real scalar field, L=−1 2∂µφ∂µφ−1 2m2φ2,(1.25) this would give 0 = δS =Zd4x−1 2∂µδφ∂µφ−1 2∂µφ∂µδφ −1 2m2φδφ=Zd4x∂µ∂µφ−m2φδφ, (1.26) where we have integrated by parts in each of the first two terms, and assumed that δφ vanishes at infinity on both the spatial and temporal dimensions, so that there is no surface term. The equation of motion resulting from (1.26) is therefore −∂µ∂µφ+m2φ= 0, which is known as the Klein-Gordon equation. Its general solution is φ(t, x) = Zd3k (2π)3/2√ωka(k)eik·x−iωkt−b(k)eik·x+iωkt,(1.27) where a(k) and b(k) are arbitrary functions, and ωk= (k2+m2)1/2. Particularizing to the case of a real field, we impose the additional condition φ(x) = φ∗(x). Using (1.27), the more general expression of a real scalar field reads φ(t, x) = Zd3k (2π)3/2√ωka(k)eikx +a∗(k)e−ikx,(1.28) where kx =k·x−ωkt. Now that we know the Lagrangian, we can also obtain the Hamiltonian following the usual procedure. Given a Lagrangian L(qi,˙qi), the conjugate momenta are given by pi=∂L ∂˙qi, and the Hamiltonian by H=Pipi˙qi−L. As we discussed before, the generalized coordinates of a classical field are φ(t, x). That is, the role of qiis played by φ(t, x). The appropriate generalizations are then Π(~x) = ∂L ∂˙ φ(x),(1.29) H= Π ˙ φ−L,(1.30) where His the Hamiltonian density, and the Hamiltonian is the spatial integral of the Hamiltonian density. Obtaining the Hamiltonian for the real scalar field is very tedious and not very enlightening, so we will only include the expression for future reference. H=1 2Zd3kωk(a∗(k)a(k) + a(k)a∗(k)).(1.31) Note that we do not add the terms inside the parenthesis. We do this in prevision of the following quantization of the real scalar field. Quantum field theory Now that we have a classical field theory, we can transform it into a quantum field theory by doing canonical quantization. This consists in upgrading the dynamical variables to operators and imposing commutation relations between them. There are several equivalent ways of doing so. In our case, we will set the commutation relations between ˆapand ˆa† q, and from here the rest of the relations can be obtained. The commutation relations between 7 the operators ˆapand ˆa† q(commonly referred to as annihilation and creation operators, respectively) are, ˆap,ˆa† q= 0, ˆa† p,ˆa† q= 0, ˆap,ˆa† q=δ(3)(p−q).(1.32) As we said before, the operators ˆa† qand ˆapare usually called creation and annihilation operators. This is because they behave in a very similar way as the creation and annihilation operators of the harmonic oscillator. Using them, we will be able to find a basis of eigenstates of the Hamiltonian, which as we remember from (1.31) is ˆ H=1 2Zd3kωk(ˆa† kˆak+ ˆakˆa† k) =Zd3kωk(ˆa† kˆak+1 2δ(3)(0)), where we have used the commutation relation (1.32). The divergence at the origin is just a constant term that can be removed by changing the energy origin. The resulting Hamiltonian will then look like ˆ H=Zd3kωkˆa† kˆak.(1.33) We now proceed to find the eigenbasis of ˆ H. We start by defining a vacuum state |0i, with the property that it is annihilated by all the ˆak, that is, ˆak|0i= 0 for all k. This state has zero energy, as clearly ˆ H|0i= 0. The rest of the eigenstates of ˆ Hare obtained by acting with the annihilation operators on the vacuum. From (1.32) and (1.33) it is easy to see that hˆ H, ˆa† ki=ωkˆa† k. Therefore, ˆ H|ki=ωk|ki, and in general ˆ H|k1, ..., kni= (ωk1+... +ωkn)|k1, ..., kni. The states of the form |k1, ..., kniform an eigenbasis of the Hilbert space of the quantum field. From now on, when we talk about free scalar fields, we will be referring to the case in which the field is additionally massless (m= 0). This will help to simplify the notation. 8 1. The expectation values hˆ A(0) ˆ B(τ)iρand hˆ B(τ)ˆ A(0)iρare boundary values of some complex functions hˆ A(0) ˆ B(z)iρand hˆ B(z)ˆ A(0)iρholomorphic in the complex plane strips 0 <Im z < β and −β < Im z < 0, respectively. 2. The following complex anti-periodicity (of period β) is satisfied by the boundary values of these complex functions: hˆ A(0) ˆ B(τ+iβ)iρ=hˆ B(τ)ˆ A(0)iρ(2.22) The inclusion of the time parameter τin the definition is not superfluous. KMS is an observer-dependent property, as well as the KMS inverse temperature β. We will now prove some results that should help to solidify the idea that KMS states are the appropriate generalization of Gibbs states. We will start by seeing that all KMS states are stationary. Applying (2.22) to the particular choice of operators ˆ A=Iand ˆ B self-adjoint, we obtain that hˆ B(τ+iβ)i=hˆ B(0)i,(2.23) which implies that hˆ B(z)iis periodic in the imaginary direction, with period iβ. Applying Schwarz’s inequality yields |hˆ B(z)i| ≤ ||ˆ B(z)|| =||ˆ B(x+iy)|| =||eiˆ Hx ˆ B(iy)e−iˆ Hx|| =||ˆ B(iy)||.(2.24) hˆ B(z)iis holomorphic, and (2.24) shows that it is bounded in the strip 0 <Imz < β. Due to its complex periodicity, hˆ B(z)iis also bounded in the rest of the complex plane. Applying Liouville’s theorem (bounded entire functions are constant), we obtain that hˆ B(z)i is constant in the complex plane. In particular, hˆ B(0)i=hˆ B(τ)i. Since ˆ Bis an arbitrary observable, this proves that a KMS state is indeed stationary. We will now prove that, when a partition function Zcan be defined, a state ˆρis a Gibbs state if and only if it is a KMS state. This is perhaps the most clear indication that KMS states are the right generalization of Gibbs states. Lets first focus on the direction Gibbs ⇒KMS. Let ˆρbe a Gibbs state of inverse temperature β. We will see that ˆρsatisfies the two KMS conditions. 1. Condition 1. The operator ˆ A(0) ˆ B(τ+iσ)ˆρ=ˆ A(0)e−σˆ Hˆ B(τ)e−(β−σ)ˆ H/Z(β),(2.25) is of trace class [25] for 0 ≤σ≤βbecause the operators on the right hand side are trace-class or bounded, and the product of a trace-class operator and a bounded operator is trace-class. As a consequence, in the complex strip 0 <Im z < β, hˆ A(0) ˆ B(z)iρ= Tr hˆ A(0) ˆ B(0)ˆρ],(2.26) is a well-defined function of z. In fact, it is a holomorphic function, as we will by checking that it satisfies the Cauchy-Riemann equations. (2.26) can be written as f(z) = Trˆ A(0)eiz ˆ Hˆ B(0)e−iz ˆ Hˆρ. Writing zas z=τ+iσ, yields ∂τf=−i∂σf. The same reasoning can be applied to hˆ B(z)ˆ A(0)i. This proves that Gibbs states satisfy condition 1. 2. Condition 2. First, note that due to stationarity, the Hamiltonian cannot be timedependent, so ˆ B(τ) = eiˆ Hτ ˆ B(0)e−iˆ Hτ .(2.27) Allowing τto take complex values we get that ˆ B(τ+iβ) = eiˆ H(τ+iβ)ˆ B(0)e−iˆ H(τ+iβ)=e−βˆ Hˆ B(τ)eβˆ H.(2.28) Substituting this and ˆρ=e−βˆ H Z(β)in (2.22), and using the cyclic property of the trace: hˆ A(0) ˆ B(τ+iβ)iˆρ= Tr hˆ A(0)e−βˆ Hˆ B(τ)eβˆ He−βˆ Zi/Z(β) = Tr hˆ B(τ)ˆ A(0)e−βˆ Hi/Z(β) = Tr hˆ B(τ)ˆ A(0)ˆρi =hˆ B(τ)ˆ A(0)iˆρ.(2.29) This completes the proof of Gibbs ⇒KMS. 15 We will now prove the converse, that KMS ⇒Gibbs when Gibbs states are welldefined. Let ˆρbe a KMS state of period β. Applying condition 2 at τ= 0 we obtain that Trˆ A(0)e−βˆ Hhˆ B(0), eβˆ Hˆρi= 0, for any bounded operator ˆ A. This implies that hˆ B(0), eβˆ Hˆρi= 0 for any bounded operator ˆ B. Therefore, eβˆ Hˆρmust be proportional to the identity ⇒ˆρis a Gibbs state. The last theorem should reassure us that KMS states are the correct generalization of Gibbs states for infinite-volume quantum systems. We will end by stating three more properties of KMS states that will be useful when working with KMS states in quantum field theory. For a KMS state ˆρβ(with inverse KMS temperature β) with respect to time translations generated by a Hamiltonian ˆ Hthe two-point correlator Wˆρ(τ, τ0):= Tr hˆρˆ φ(t(τ)x(τ)) ˆ φ(t(τ0)x(τ0))isatisfies the following conditions: 1. Wρ(τ, τ0) = Wρ(∆τ) (Stationarity). 2. Wρ(∆τ+β) = Wρ(−∆τ) (C−antiperiodicity). 3. Holomorphicity of the two-point correlator Wˆρ(0, z) in the upper complex strip 0 < Im z < β. If we want to prove Jarzynski and Crooks theorems for non-equilibrium processes occurring in quantum fields with full generality, we will have to do it for the case in which the initial state of the system is a KMS state, and not simply a Gibbs state. One last problem that we should solve is how to define a work distribution in quantum field theory. Given the success that the TPM scheme has had reproducing fluctuation theorems in finite-dimensional quantum systems, it seems natural to also use the TPM scheme to define work distributions in quantum field theory. However, there is a problem with this approach. As we will explain in more detail in the next chapter, projective measurements cannot be allowed in a relativistic quantum theory, so readily applying the TPM scheme to quantum fields is not possible. We will explain how this problem is avoided in relativistic quantum information, and we will draw inspiration from here to define a work distribution using the Ramsey scheme introduced in this chapter. 16 Chapter 3 Relativistic Quantum Information The focus of this chapter will be to introduce a way of measuring quantum fields that has been extensively used in the last years to show a wide variety of phenomena, such as entanglement harvesting [26, 27], communication through scalar fields [28]... and that is consistent with relativity. We will give two examples of how this formalism is applied: we will show that there is a non-zero probability of a detector jumping from the ground to the excited state when it is in contact with the vacuum state of a quantum field. We will also show that a detector that is in contact for a sufficiently long time with a quantum field in a KMS state will end up being in a thermal state. This is another proof that KMS states are the correct generalization of Gibbs states. 3.1 Measuring quantum fields The question of how to measure a quantum field, was shown to be much more subtle than expected by Sorkin in [29]. He showed that, if we are allowed to do even localized projective measurements in a quantum field, we also have the possibility of doing faster than light communication. The way this is proven is by choosing three regions of spacetime O1, O2, O3, so that some points of O1are in the causal past of some points of O2, and some points of O2are in the causal past of some points of O3, but O1and O3are spacelike separated. For general observables A, B, C measured in the regions O1, O2, O3, respectively, and for a general state of the field ˆρ, it can be proven that, if Bis measured, the result of Cdepends on whether Awas measured, even though Aand Care spacelike separated. Therefore, by arranging that Bwill be measured with certainty, someone at O1could use this dependency of Con Ato transmit information superluminally to O3. This result raised the question of which type of measurements could be done in quantum fields. A definite answer to this question was recently given by [30] in the context of algebraic quantum field theory, which is far beyond the scope of this text. The main idea is that measurements on a quantum field are made by interacting with the field on a finite region of spacetime using a probe, and measuring the probe afterwards. This technique had been used frequently in Relativistic Quantum Information, and many interesting effects such as entanglement harvesting [26, 27], communication through scalar fields [28], were studied using this procedure. Usually (although not always [31]), the quantum field to be measured will be a scalar field. In this case, it is customary to use as a probe a two-level system, whose interaction Hamiltonian with the field is given by (in the Dirac picture) ˆ HUDW (t) = λχ(t)Zd3xF(x)ˆµ(t)ˆ φ(t, x).(3.1) ˆµ(t) = σ+eiΩt+σ−e−iΩtis the monopole operator, and F(x) and χ(t) are the smearing and switching functions respectively. F(x) represents the geometric shape of the two level system (also referred to as detector), and χ(t) the intensity and the duration of the coupling. Ω is the energy gap of the two-level system and σ+is the analogue of the ˆσxoperator (ˆσ−= ˆσ+†). This model of interaction is known as the Unruh-DeWitt model, and is inspired in the light-matter interaction. 17 The main message to be taken away from this section is that quantum fields are measured using detectors. This is something that we will have to keep in mind when defining a notion of work distribution for quantum field theory. Such definition should not be based in the results of projective measurements directly applied on the field, but on the results collected when measuring detectors that have interacted with it. We will now move on to prove that an Unruh-DeWitt detector that starts in its ground state |gican be found with non-zero probability to be in its excited state |eiafter interacting with the vacuum state of a scalar field. Example 1: Excitations of UDW detectors The Hamiltonian of the field and detector system is ˆ H=ˆ Hf+ˆ HT LS +ˆ HUDW ,(3.2) where ˆ Hfis the Hamiltonian of the free scalar field (1.12), ˆ HT LS is the Hamiltonian of the two level system, ˆ HT LS =Ω 2ˆσZand ˆ HUDW is the Unruh-DeWitt Hamiltonian (3.1) that mediates the interaction between the two level system and the field. We will assume that λis small, so that we can use perturbation theory in our calculations. We want to obtain P|0i |gi→|ei(Ω), namely the probability of the two level system of going from the ground to the excited state when it is in contact with the vacuum state of the free scalar field, as a function of the energy gap Ω. Clearly, P|0i |gi→|ei(Ω) = X out |hout, e|ˆ U|g, 0i|2=X out h0, g|ˆ U†|e, outihout, e|ˆ U|g, 0i,(3.3) where ˆ U=Texp R+∞ −∞ ˆ HUDW (t)dt, and the sum is done over all the possible final states of the field. Expanding ˆ Uin power series as presented in (1.21), we obtain that P|0i |gi→|ei(Ω) = X out h0, g|I+ˆ U†(1) +O(λ2)|e, outihout, e|I+ˆ U(1) +O(λ2)|g, 0i =X out h0, g|ˆ U†(1) |e, outihout, e|ˆ U(1) |g, 0i+O(λ3).(3.4) Here, we have used that the excited and ground states of the detector are orthogonal. Substituting the expression for ˆ U(1) (1.22), and for ˆ HUDW (t) (3.1) (remembering that the field ˆ φ(x) is a free scalar field (1.28)) yields P|0i |gi→|ei(Ω) = λ2Z+∞ −∞ dtZ+∞ −∞ dt0χ(t)χ(t‘) hg|ˆµ(t)|eihe|ˆµ(t‘) |giZd3xZd3x0h0|ˆ φ(t, x)ˆ φ(t0,x0)|0i. (3.5) We now use the expressions of hg|ˆµ(t)|ei=e−iΩtand the two-point correlator of the field h0|ˆ φ(t, x)ˆ φ(t0,x0)|0i=Zd3k 2(2π)3ωk e−i(ωk(t−t0)−k·(x−x0)),(3.6) to obtain for the excitation probability of the detector P|0i |gi→|ei(Ω) = λ2Zd3k 2(2π)3ωk|eχ(Ω + ωk)|2|e F(k)|2,(3.7) where eχand e Fare the Fourier transforms of the switching and smearing functions. As we can see, there is a non-zero probability for the detector to jump to the excited state. We will now pause for a moment to explain why this result is counter-intuitive. The Unruh-DeWitt model is a simplified model for the light matter interaction. However, for certain physical scenarios, such as those considered in quantum optics, this model can still be unnecessarily complex. In theses cases, usually, two further assumptions are performed: the single mode approximation and the rotating wave approximation [32]. When these approximations are made, the final Hamiltonian for the light-matter interaction is that of the Jaynes-Cummings model ˆ HJC =λ(ˆσ+ˆaΩ+ ˆσ−ˆa† Ω).(3.8) 18 The interpretation of this Hamiltonian is clear. The two level system can go from the ground to the excited state by absorbing a photon whose energy equals its energy gap Ω (this is the term ˆσ+ˆaΩ). Conversely, if the two level system starts in the excited state, it can deexcite emitting a photon in the process. As we can see, with this model of interaction, it is impossible that the two-level system goes from the ground to the excited state if it interacts with a field that is in its vacuum state (no photons), as opposed to what we see if we use the Unruh-DeWitt detector model. This is just an example of a series of features of the light-matter interaction that appear when we use the UDW model, but not under certain additional approximations [33]. We will now see another example of how to use detectors to establish properties of quantum fields. Example 2: Thermalization of an UDW detector We will show that, under certain conditions, an Unruh-DeWitt detector that interacts for a long time with a KMS state of a scalar field, will end up being in a thermal state. In order to prove this, we will show that Pexc(Ω) Pdeexc(Ω) =e−βΩ,(3.9) where βis the inverse temperature of the KMS state of the field, and Pexc(Ω) and Pdeexc(Ω) are the excitation and deexcitation probabilities of the detector when it is in contact with the field. For simplicity, we will assume that the detector is pointlike, that is F(x) = δ(3)(x). Also, we will introduce a timescale σ, that will control the duration of the interaction. Particularizing (3.5) to the case of a point-like detector, we have that P(Ω) = λ2σF(Ω, σ), where F(Ω, σ) = 1 σZ+∞ −∞ dτZ+∞ −∞ dτ0χτ σχτ0 σeiΩ(τ−τ0)W(τ, τ0),(3.10) Wˆρ(τ, τ0) = Tr hˆρˆ φ(t(τ)x(τ)) ˆ φ(t(τ0)x(τ0))iand ˆρis the state of the field. Now we write χ(τ σ) as χτ σ=1 2πZ+∞ −∞ dωeχ(ω)e−iω τ σ,(3.11) and χτ σ=1 2πZ+∞ −∞ dωeχ∗(ω)eiω τ σ.(3.12) Therefore, F(Ω, σ) = 1 4π2σZ+∞ −∞ dτZ+∞ −∞ dτ0Z+∞ −∞ dωZ+∞ −∞ dω0eχ∗(ω)eχ(ω0)ei(ωτ σ−ω0τ0 σ)W(τ, τ0)ei(τ−τ0)Ω. (3.13) Since the field is in a KMS state with respect to τ, it is also stationary with respect to τ,W(τ, τ0) = W(τ−τ0). By doing the change of variables u=τ−τ0 v=τ+τ0(3.14) we can rewrite (3.13) as F(Ω, σ) = 1 8π2σZ+∞ −∞ duZ+∞ −∞ dvZ+∞ −∞ dωZ+∞ −∞ dω0eχ∗(ω)eχ(ω0)ei(ωu 2σ+ω0u 2σ)ei(ωv 2σ−ω0v 2σ)W(u)eiuΩ =1 2πZ+∞ −∞ duZ+∞ −∞ dω|eχ(ω)|2W(u)ei(Ω+ ω σ)u =1 2πZ+∞ −∞ dω|eχ(ω)|2f WΩ + ω σ.(3.15) 19 We want the limit of long interactions (large σ). If eχ(ω) decays fast enough, we arrive to lim σ→∞ F(Ω, σ) = 1 2πZ+∞ −∞ dω|eχ(ω)|2f W(Ω) = f W(Ω) 2πZ+∞ −∞ dω|eχ(ω)|2=f W(Ω).(3.16) Now, using the C−antiperiodicity of the two-point correlator of KMS states W(∆τ+iβ) = W(−∆τ)⇒Z+∞ −∞ d∆τW(∆τ+iβ)eiω∆τ=Z+∞ −∞ d∆τW(−∆τ)eiω∆τ. (3.17) The RHS is equal to f W(−ω). For the LHS, we do the change of variables ∆τ0= ∆τ+iβ and integrate ver the Im z=iβ line: Z+∞ −∞ d∆τW(∆τ+iβ)eiω∆τ=Zγ d∆τ0W(∆τ0)eiω(∆τ0−iβ)=eβω Zγ d∆τ0eiω∆τ0=eβω f W(ω). (3.18) For the last step we have used the holomorphicity of the two-point correlator Wˆρ(0, z) in the upper complex strip 0 <Im z < β and Cauchy integral theorem. Putting everything together, we have that Pexc(Ω) Pdeexc(Ω) =F(Ω, σ) F(−Ω, σ)=f W(Ω) f W(−Ω) =e−βω,(3.19) as desired. With this we conclude our section of examples on how to use detectors to study nontrivial properties of quantum fields. In the next chapter, we will use this idea to defnie a notion of work distribution for quantum fields, that will satisfy Crooks and Jarzynski theorems. 20 Chapter 4 Work distributions on quantum fields In this chapter we will summarize the results obtained in [1]. The complete article is included in the Appendix. Our objective is to extend Crooks and Jarzynski theorems to non-equilibrium processes occurring in quantum fields. There are two main difficulties for doing so. First, if we want our results to be general, we will have to work with KMS states, and not simply with Gibbs states. Second, we cannot readilty extend the TPM scheme to quantum field theory, as it involves doing projective measurements of a quantum field. We have to define a work distribution based on the results of measurements done in detectors. In Chapter 2 we introduced the Ramsey scheme, as a way of experimentally obtaining the work distribution for processes occurring in finite-dimensional quantum systems. It involved a controlled interaction of a qubit and the system of interest, and later by measuring the state of the qubit the characteristic function of the work distribution could be obtained. This is precisely the type of operational schemes that we need to define a work distributions in quantum fields avoiding projective measurements, and is actually the way we define it. In [1] we operationally define work distributions of quantum fields through the Ramsey scheme, and then start studying its properties. We study work distributions of processes whose Hamiltonians are of the form ˆ Hφ(t) = ˆ H0+λχ(t)ZR3 d3xF(x)ˆ φ(t, x) = ˆ H0+ˆ HI(t),(4.1) in the interaction picture, where ˆ H0is the free Hamiltonian of the field (1.33). We assume that the switching function has strong support in a finite region and, without loss of generality, we take the strong support of the switching function to be in the interval [0, T], where 0 and Tare the starting and ending times of the process under study. Assuming that the coupling constant λis small, we can use perturbation theory to obtain closed expressions for the characteristic function of the work distribution to second order in λ, for an arbitrary KMS state of the massless free scalar field: e P(µ) = 1 + λ2Zd3k (2π)32ωk(eβωk−1)|eχ(ωk)|2|e F(k)|2×eβωk+ 1(cos(µωk)−1) +iλ2Zd3k (2π)32ωk|eχ(ωk)|2|e F(k)|2sin(µωk).(4.2) Now that we have a closed expression for the characteristic function, we will try to prove Jarzynski and Crooks theorems. Since we are considering cyclic processes ( ˆ H(0) = ˆ H(T)), Jarzynski equality reads in this case he−βW i= 1. By evaluating e P(µ) at iβ we obtain the desired result. Now, to prove Crooks theorem we have to do the inverse Fourier transform of (4.2) to recover the work probability distribution P(W) P(W) = (1 −p)δ(W) + λ2 2π|eχ(W)|2|e F(W)|2W×eβW eβW −1Θ(W) + 1 1−e−βW Θ(−W), (4.3) 21 where p:=RW6=0 dW P(W) and Θ(W) is the Heaviside function. It is just a calculation to check that P(W) Prev(−W)=eβW . With this we have proved that for unitaries generated by Hamiltonians of the form (4.1), our definition of work distribution satisfies Jarzynski and Crooks theorems. This, together with the definition of P(W) for quantum fields through the Ramsey scheme, are the two main results of [1]. We also study some features of the work probability distribution (4.3). Of significant interest is to understand when the work fluctuations dominate the work average values. From the characteristic function (4.2) it is possible to obtain by differentiation the different moments of P(W). We find that, for the vacuum (β=∞) hWi=λ2Zd3k (2π)32|eχ(ωk)|2|e F(k)|2,(4.4) σ2 β=λ2 2Zd3k (2π)3 eβωk+1 eβωk−1|eχ(ωk)|2|e F(k)|2ωk+O(λ4).(4.5) An interesting observation is that, for the vacuum, if we consider unitaries that are very localized in time and space, both eχ(ωk) and e F(k) will be wide in the frequency space, which means that the work variance will become larger than the expectation value, making the variance of the work increasingly significant as the operation on the field becomes increasingly localized in both time and space. Several other observations about the work distribution (4.3) can be made. We refer the reader to [1] for a deeper analysis. Future work could involve use our definition of work distribution to study the thermodynamics of processes of interest occurring in quantum fields, such as entanglement harvesting, quantum energy teleportation [34]... 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Impossible measurements on quantum fields. arXiv:gr-qc/9302018v2, 1993. [30] Rainer Verch Christopher J. Fewster. Quantum fields and local measurements. arXiv:1810.06512v1[math-ph], 2018. [31] Alejandro Pozas-Kerstjens and Eduardo Mart´ın-Mart´ınez. Entanglement harvesting from the electromagnetic vacuum with hydrogenlike atoms. Phys. Rev. D, 94:064074, Sep 2016. doi: 10.1103/PhysRevD.94.064074. URL https://link.aps.org/doi/10.1103/PhysRevD.94.064074. 24 6 only difference being the presence of a factor e−iµ ˆ H0multiplying the ket vectors |ki. Since e−iµˆ H0|ki=e−iµωk|ki, we obtain that Tr e−iµˆ H0ˆ U(1) |ΩihΩ|eiµˆ H0ˆ U†(1)=λ2Zd3k (2π)32ωk|eχ(ωk)|2|e F(k)|2e−iµωk.(21) The rest of the components are the Hermitian conjugates of these. We now calculate the remaining terms. That is, the terms that involve products with ˆ U(2). Let us start by obtaining Trφˆ U(2)e−iµˆ H0|ΩihΩ|eiµˆ H0⊗|0ih0| 2+H.c=hΩ|ˆ U(2) |Ωi|0ih0| 2+H.c. This is simply −λ2Z∞ −∞ dtZt −∞ dt0Zd3xZd3x0χ(t)χ(t0)F(x)F(x0)hΩ|ˆ φ(t, x)ˆ φ(t0,x0)|Ωi|0ih0| 2+H.c (22) =−λ2Z∞ −∞ dtZt −∞ dt0Zd3xZd3x0χ(t)χ(t0)F(x)F(x0)2 Re[W(t, x, t0,x0)]|0ih0| 2, where W(t, x, t0,x0) = hΩ|ˆ φ(t, x)ˆ φ(t0,x0)|Ωiis the Wightman function. The same is obtained for the other cases. This is because all the e±iµ ˆ H0end up multiplying the vacuum state when taking the trace, so they disappear leaving simply hΩ|ˆ U(2) |Ωi+H.c. Therefore, the contribution of these terms to the reduced state of the qubit is A|+ih+|, where A=−λ2Z∞ −∞ dtZt −∞ dt0Zd3xZd3x0χ(t)χ(t0)F(x)F(x0)2 Re[W(t, x, t0,x0)].(23) Adding everything and noting that (20) is equal to (23) we obtain, after applying the second Hadamard on the qubit, that the reduced state can be written as ˆρµ=1 2I+|−ih+|1 + λ2Zd3k (2π)32ωk|eχ(ωk)|2|e F(k)|2e−iµωk−1(24) +|+ih−|1 + λ2Zd3k (2π)32ωk|eχ(ωk)|2|e F(k)|2eiµωk−1.(25) Field in a finite-temperature KMS state Some properties that we use throughout these calculation are: Trˆ φˆρβ= 0,(26) hˆρβ, e−it ˆ H0i= 0.(27) As before, we calculate the reduced state of the qubit with a Dyson expansion. The first order term is again zero. As an example of why this is the case, we calculate Tr e−iµˆ H0ˆ U(1) ˆρβeiµˆ H0+ Tr e−iµˆ H0ˆρβeiµˆ H0ˆ U†(1)= Tr ˆ U(1) ˆρβ+Tr ˆρβˆ U†(1),(28) where we have used the cyclic property of the trace and (27). Finally, using the linearity of the trace, the expression for ˆ U(1) and (26), we obtain that both terms are zero. A similar procedure can be used to check that all the other contributions to the first order correction are zero. We now calculate the second order terms, starting by the ones that only involve products of ˆ U(1) and ˆ U†(1). We derive here only the coefficient of |0ih1| 2. The other cases follow analogously. Tr ˆ U(1)e−iµˆ H0ˆρβˆ U†(1)eiµˆ H0= Tr eiµˆ H0ˆ U(1)e−iµˆ H0ˆρβˆ U†(1).(29) We have that eiµˆ H0ˆ U(1)e−iµˆ H0=−iλZ∞ −∞ dtZd3xχ(t)F(x)eiµˆ H0ˆ φ(x, t)e−iµˆ H0(30) =−iλZ∞ −∞ dtZd3xχ(t)F(x)ˆ φ(x, t +µ). 7 So (29) equals λ2ZdtZdt0Zd3xZd3x0χ(t)χ(t0)F(x)F(x0) Tr ˆ φ(x, t +µ)ˆρβˆ φ(x0, t0+µ)(31) =λ2ZdtZdt0Zd3xZd3x0χ(t)χ(t0)F(x)F(x0)Wβ(x0, t0,x, t +µ). The other terms have the same structure, with the only change being in the thermal Wightman function, which is W(x0, t0, x, t) for the diagonal terms, and W(x0, t0, x, t −µ) for the |1ih0| 2term. Let us obtain now the second order terms coming from products with ˆ U(2) and ˆ U†(2). It is easy to see, using (27) that all the terms are equal to Tr ˆ U(2) ˆρβ+ Tr ˆρβˆ U†(2)= 2 Re Tr ˆ U(2) ˆρβ.(32) This finishes the proof. Since the Dyson expansion preserves the trace of the density matrix, Eq. (31) has to be equal to Eq. (32), so as to cancel the diagonals added by the perturbation terms. This is useful because it gives a much more compact expression for the reduced state of the qubit. Therefore, at the end of the Ramsey scheme, the density matrix of the qubit is: 1 2I+|+ih−|1 + λ2ZdtZdt0Zd3xZd3x0χ(t)χ(t0)F(x)F(x0) (Wβ(x0, t0,x, t +µ)−Wβ(x0, t0,x, t))(33) +|−ih+|1 + λ2ZdtZdt0Zd3xZd3x0χ(t)χ(t0)F(x)F(x0) (Wβ(x0, t0,x, t −µ)−Wβ(x0, t0,x, t)). 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