RESUMEN ANALÍTICO EN EDUCACIÓN RAE
6. Conclusiones Las conclusiones dadas en este documento son:
The research presented in this thesis was pursued to push further the idea of circuit QED with multiple components, both with regard to quantum optics and quantum simulations, while being experimentally realistic in its proposals. It is concerned with two main topics.
First, we consider the simplest possible multi-qubit circuit QED scenario, the equilibrium behavior of many superconducting charge qubits coupled to a cavity. We re-examine the possi- bility of superradiant phase transitions in such a system. Our study is motivated by the following observation. On the basis of standard theory of circuit QED systems, which was introduced in the previous sections of this chapter, one is lead to the conclusion that superradiant phase transi- tions should be in principle observable in circuit cavity QED systems. However, for realatoms coupling to a bosonic mode, superradiant phase transitions cannot occur. Real atoms are subject to a no-go theorem discovered byRza˙zewskiet al.(1975), only shortly after superradiant phase transitions had been first discussed (Hepp and Lieb, 1973). This would mean that, in this one aspect, the otherwise well-established similarity of circuit cavity QED systems and atomic cavity QED systems fails.
In Chapter3, we resolve this problem by employing a fundamental, microscopic description of the considered circuit QED system. This enables us to apply the no-go theorem to circuit QED systems as well and, thus, to reject the possibility of superradiant phase transitions in circuit QED systems with charge qubits. Although the standard description of circuit QED systems
has proved to be highly useful in many situations, and is certainly more convenient than our microscopic description, we may also conclude that the standard description can lead to even qualitatively incorrect predictions and can no longer be trusted when proceeding to large-scale circuit QED systems. In addition to that, we scrutinize the no-go theorem and generalize it to multi-level atoms so as to make it more applicable to realistic systems. Sections 3.1 - 3.3
provide detailed introductions to superradiant phase transitions, the no-go theorem, and previous discussions of this issue in the context of circuit QED, respectively. Our results on this topic have been published in a research article, which is reprinted in Section3.4.
Second, we propose a circuit QED setup that simulates the quantum Ising chain in a time- dependent transverse magnetic field. In particular, we argue that our setup is suited for observing the non-equilibrium dynamics of the transverse-field Ising chain. Our proposal is motivated as follows. Non-equilibrium quantum physics is currently subject to much theoretical research. However, experimental platforms facilitating its time-resolved observation are rare and so far es- sentially limited to systems of cold atoms in optical lattices. Our proposal might help to alleviate this shortage. Moreover, based on a flexible design, its implementation might be an important benchmark for future circuit QED quantum simulators of non-integrable quantum many-body spin systems, whose dynamical behavior can no longer be predicted by a classical calculation. To measure the behavior of such quantum simulators and to obtain in that way the solutions of computationally intractable problems is the central goal of all quantum simulations. Compared to the circuit QED quantum simulators we have discussed previously in this section, the pro- posed system relies on a different and possibly simpler concept – the direct capacitive coupling of charge qubits – so that first results on its experimental implementation have already been obtained, as described below.
In Chapter4, we introduce this setup, derive its Hamiltonian, and calculate its spectrum to fa- cilitate its initial experimental characterization. Experiments for observing the non-equilibrium dynamics of the transverse-field Ising chain are suggested. The expected behavior is calculated for typical circuit QED parameters and interpreted using the tools developed in previous work on non-equilibrium physics. A systematic study of the influence of disorder on these experiments is provided. Small amounts of fabrication-induced disorder are shown not to spoil the predicted experimental results. Engineering the Ising chain with a larger degree of disorder would allow the study of interesting new effects such as Anderson localization of propagating excitations. We also describe the experimental implementation of our proposal, which we have pursued in collab- oration with the group of Professor Irfan Siddiqi at UC Berkeley. We present preliminary results in this regard and compare them with our theory. Sections 4.1 and 4.2 provide a motivation of and an introduction to quantum simulations, with an emphasis being placed on the impor- tance of simulating non-equilibrium systems. Arguments for pursuing quantum simulations in circuit QED are brought forward in Section4.3. Some properties of the quantum Ising chain are summarized in Section 4.4, and its importance is highlighted. Reprints of our previously pub- lished research articles on the simulation of the non-equilibrium dynamics of the transverse-field Ising chain in circuit QED and on the influence of disorder on such simulations are contained in Sections4.5 and4.6, respectively. The experimental realization of our proposal is discussed in Section4.7.
tum technologies for implementing quantum protocols such as circuit QED but also intense fun- damental investigations on the impact of quantum physics on information theory. The quantifica- tion and the classification of entanglement are an important part of this endeavour and formed a side project to the work presented in this thesis. The results obtained in this context are presented inViehmannet al.(2011,2012b).
Superradiant phase transitions in
circuit QED
3.1
Superradiant phase transitions
If, instead of a single two-level atom, N atoms couple resonantly to the electromagnetic field in a cavity, the splitting of the two excited states with a single excitation shared between cavity and atomic ensemble will increase in proportion to√N. This follows from the Tavis-Cummings model. Clearly, if the model remains valid, at some large N one of these excited state will no longer be separated by an energy gap from the ground state of the system, and a phase transi- tion occurs.1 This phase transition, which survives up to some nonzero temperature and is not tied to resonant atom field coupling, was first investigated byHepp and Lieb(1973) and termed
superradiant phase transition (SPT) because the occupied states in the new phase possess the potential to superradiate. A mathematically less rigorous but greatly simplified treatment was provided byWang and Hioe(1973). The SPT persists if atoms and cavity are off-resonant and if the counter-rotating terms are included in the Hamiltonian, which cannot be neglected in this regime of strong coupling (Carmichaelet al., 1973). With counter-rotating terms, the Hamilto- nian considered by these authors reads (from now on, we set~=1)
H
D,0=ωa†a+ Ω 2 N∑
j=1 σzj+√λ N N∑
j=1 σxj(a†+a), (3.1)which is usually calledDicke Hamiltonianin the context of superradiance and superradiant phase transitions. The notation is the same as for the Jaynes-Cummings Hamiltonian (2.25) except for
g→λ/√N(see below) andωc→ω(to avoid confusion with critical quantities) throughout this chapter. The system is considered in the thermodynamic limit N,V →∞with constant particle density N/V, whereV is the quantization volume of the electromagnetic field. Therefore, it is convenient to pull out a factor of 1/√Nfrom the coupling terms so that the coupling parameterλ (we adopt here the common notation) is a well-behaved function of the particle density N/V.
Explicitly, λ ∝ Ω|e·d|pN/V. Here,d=he|∑ni=1qiri|giis the dipole matrix element between the kept two atomic levels {|gi,|ei} of the atoms, which are assumed to be identical and to consist of n particles with charge qi and coordinate ri, and e is the polarization vector of the vector potential A. The Dicke Hamiltonian can be derived from Equations (2.26) and (2.27) under the same approximations as the Jaynes-Cummings Hamiltonian, and with the additional assumption that theN atoms are located in an area that is small compared to the wave length of the one considered field mode.
By direct evaluation of the system’s partition function, it was shown that for λ> λc,0 = √
ωΩ/2, there is a critical temperatureTc at which a second-order phase transition occurs. This temperature satisfies tanh Ω 2kBTc = ωΩ 4λ2, (3.2)
where kB is the Boltzmann constant. For λ<λc,0, the system does not become critical. In the ‘normal phase’, characterized by λ<λc,0 orλ>λc,0 andT >Tc, there is no macroscopic population of the boson mode, ha†ai/N=0. In the ‘superradiant phase’ (λ>λc,0 andT <Tc), the bosonic mode is macroscopically occupied,
ha†ai N =4 λ2 ω2x 2 − Ω 2 16λ2, (3.3)
wherexsolves 2x=tanh(4λ2x/(ωkBT))>0.
In recent years, there has been much interest in the phase transition of the Dicke model, largely becauseEmary and Brandes(2003a,b) found exact results for the eigenvalues and eigen- states of the low-energy sector of
H
D,0 in the limit N,V →∞. This was achieved by deriving effective Hamiltonians forH
D,0for the casesλ≶λc,0 in a self-consistent procedure: For weak coupling, it is justified to assume that the low-lying states of the Hamiltonian do not exhibit a macroscopic population of the excited atomic levels, h∑jσj
zi/N =−1. Thus, the atomic en- semble can be regarded as an effective harmonic oscillator since one can arbitrarily often excite a randomly chosen atom of the ensemble with one and the same quantum of energy without ever finding an atom already being in its excited state. Mathematically, this deliberation can be expressed by applying a Holstein-Primakoff transformation (Holstein and Primakoff, 1940) in Equation (3.1),
∑
j σ+j √ N =b † r 1−b †b N ,∑
j σ−j √ N = r 1−b †b N b,∑
j σzj 2 =b †b−N 2, (3.4)where b is bosonic, and dropping all terms with N in the denominator (Emary and Brandes,
2003a,b).2 Equivalently, one can apply a Hopfield transformation(Hopfield, 1958), which we 2. We remark that Equations (3.4) presume that the atomic ensemble is in a state with maximum pseudo angular
0 0.5 1 1.5 0.5 0.25 0 0 0.5 1 1.5 0.5 0.25 0 0 0.5 1 1 0.5 0.75 0.25 0 0.5 1 1 0.5 0.75 0.25 0 0.5 1 1.5 1 2 0 0 0.5 1 1.5 1 2 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 3.1: Properties of the Dicke Hamiltonian in the thermodynamic limitN,V →∞. For all plots, resonant atom-field coupling was assumed,ω=Ω. (a) Excitation energiesεvs. couplingλ. Atλc=
√
ωΩ/2, the second-order superradiant phase transition occurs, indicated by the dashed line (only in this panel). The branches ε≶± are the eigenmodes of the effective Hamiltonians for the low-energy sectors of the Dicke Hamiltonian in the two phases. The inset shows the scaled ground state energy Eg/N vs.λ, which is nonanalytic at the phase transition. (b) Scaled occupation of the bosonic mode,ha†ai/N, vs. couplingλ. Inset, excitation probabilityP↑of one specific atom vs.λ.
will make extensive use of (see later). Diagonalization of the resulting Hamiltonian gives the scaled ground state energyEg</N=−Ω/2 and the eigenmodesε<±,
2(ε<±)2=ω2+Ω2±
q
(ω2−Ω2)2+16λ2ωΩ, (3.5)
which fully describe the low-energy spectrum of the Dicke Hamiltonian in the normal phase (Figure 3.1(a)). In particular, one realizes that one eigenmode of the system becomes gapless at λ=λc,0. At this point, the assumption of dilute atomic excitations breaks down. However, if one assumes instead of dilute atomic excitations that the photon field a and the Holstein- Primakoff field b are macroscopically displaced, one finds a new effective Hamiltonian for the low energy sector of
H
D,0 valid for λ>λc,0. Diagonalization yields the scaled ground state energyEg>/N=−λ2/Ω+ωΩ2/(16λ2)and excitation energies2(ε>±)2=ω2+Ω2/µ2±
q
(ω2−Ω2/µ2)2+4ω2Ω2, (3.6) whereµ=ωΩ/(4λ2)(Figure3.1(a)). In fact, there are two effective Hamiltonians forλ>λc, which are identical but result from two different possibilities to displace the fieldsaandb. This reflects the broken parity symmetry of
H
D,0 (which couples only states with an even or odd number of excitations, respectively) in the superradiant phase. Note also that the all states aredoubly degenerate for λ>λc,0, a consequence of the broken symmetry. The probability P↑ of finding an atom in an excited state, P↑=h∑jσ
j
zi/(2N) +1/2, and the macroscopic occupation of the photon mode,ha†ai/N, are given by
P↑= 1 2− ωΩ 8λ2, ha†ai N = λ2 ω2− Ω2 16λ2 for λ>λc,0, (3.7)
and both are zero forλ<λc,0 (see Figure3.1(b)). The RHSs of Equations (3.7) are the squared displacements of the fields b and a divided by N. We remark that the displacements increase ∝pλ−λc,0close to the phase transition and can be regarded as order parameters. This approach allowedEmary and Brandes(2003a,b) to study in a very detailed way the zero-temperature phase transition of the Dicke Hamiltonian, which we will also focus on in the following.