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Optics and Spectroscopy

Evolution of microwave quantum states in terms of measurable ordered moments of creation and annihilation operators

Sergey N. Filippov, Vladimir I. Man’ko

Published Mar 1, 2012• 2 citations
Publisher

Abstract

The detection of microwave states is complicated by strong thermal noise, which is inevitably introduced by linear amplifiers. We show how to extract from measured data normally or anti-normally ordered moments of photon creation and annihilation operators, the set of which contains complete information on the quantum state of an electromagnetic field. Equations for the evolution of the quantum state are derived in terms of moments. Using this approach, we consider in detail issues of decoherence and thermalization of microwave quantum states. Results are illustrated using the examples of Fock, coherent, squeezed, thermal, and even and odd coherent states (Schrödinger cat states).

DOI
10.1134/s0030400x12030083

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The detection of microwave states is complicated by strong thermal noise, which is inevitably introduced by linear amplifiers. We show how to extract from measured data normally or anti-normally ordered moments of photon creation and annihilation operators, the set of which contains complete information on the quantum state of an electromagnetic field. Equations for the evolution of the quantum state are derived in terms of moments. Using this approach, we consider in detail issues of decoherence and thermalization of microwave quantum states. Results are illustrated using the examples of Fock, coherent, squeezed, thermal, and even and odd coherent states (Schrödinger cat states).
Two measurable characteristics of microwave one-mode photon states are discussed: a rotated quadrature distribution (tomogram) and normally and antinormally ordered moments of photon creation and annihilation operators. Extraction of these characteristics from an amplified microwave signal is presented. Relations between the tomogram and the moments are found and can be used as a cross-check of experiments. Formalism of the ordered moments is developed. The state purity and generalized uncertainty relations are considered in terms of moments. Unitary and nonunitary time evolution of moments is obtained in the form of a system of linear differential equations in contrast to partial differential equations for quasidistributions. Time evolution is specified for the cases of a harmonic oscillator and a damped harmonic oscillator which describe noiseless and decoherence processes, respectively.
A cat-state is a superposition of two coherent states with amplitudes α0α_{0}α0​ and −α0-α_{0}−α0​. Recent experiments create cat states in a microwave cavity field using superconducting circuits. As with degenerate parametric oscillation (DPO) in an adiabatic and highly nonlinear limit, the states are formed in a signal cavity mode via a two-photon dissipative process induced by the down conversion of a pump field to generate pairs of signal photons. The damping of the signal and the presence of thermal fluctuations rapidly decoheres the state, and the effect on the dynamics is to either destroy the possibility of a cat state, or else to sharply reduce the lifetime and size of the cat-states that can be formed. In this paper, we study the effect on both the DPO and microwave systems of a squeezed reservoir coupled to the cavity. While the threshold nonlinearity is not altered, we show that the use of squeezed states significantly lengthens the lifetime of the cat states. This improves the feasibility of generating cat states of large amplitude and with a greater degree of quantum macroscopic coherence, which is necessary for many quantum technology applications. Using current experimental parameters for the microwave set-up, which requires a modified Hamiltonian, we further demonstrate how squeezed states enhance the quality of the cat states that could be formed in this regime. Squeezing also combats the significant decoherence due to thermal noise, which is relevant for microwave fields at finite temperature. By modeling a thermal squeezed reservoir, we show that the thermal decoherence of the dynamical cat states can be inhibited by a careful control of the squeezing of the reservoir. To signify the quality of the cat state, we consider different signatures including fringes and negativity, and the Cl1C_{l_{1}}Cl1​​ measure of quantum coherence.
In this chapter, we will introduce different representations of the electromagnetic (EM) field. One may ask, why do we need different representations for the EMfield? The answer is that usually we do not know the state of the EM field which in quantum optics is represented by the annihilation and creation operators. However, results of measurements of the field are given in terms of real variables, for example, intensity. The representations allow us to recognize the state of the field from values of the measured quantities such as average amplitude, intensity, and correlation functions. The nature of the state is present in terms of the interpretation of the apparently classical (measured) variables. We will discuss two basic types of representations often used in quantum optics, Fock state (photon number) representation and coherent states representation, the later one introduced by Glauber.a We also discuss properties of fields with thermal and Poisson distribution of photons. The photon number states are very often used as a basis for quantum optics problems, and despite of many difficulties have recently been generated experimentally [11-13].
The photon creation and annihilation operators are cornerstones of the quantum description of the electromagnetic field. They signify the isomorphism of the optical Hilbert space to that of the harmonic oscillator and the bosonic nature of photons. We perform complete experimental characterization (quantum process tomography) of these operators. By measuring their effect on coherent states by means of homodyne tomography, we obtain their process tensor in the Fock basis, which explicitly shows the "raising" and "lowering" properties of these operators with respect to photon number states. This is the first experimental demonstration of complete tomography of nondeterministic quantum processes.
It has been argued that the positive-frequency part of the quantized electromagnetic field is the "observable" that one would most naturally associate with field measurements using quantum photodetectors. However, since it is possible in principle to make field measurements via the process of stimulated emission, the question of the possible solutions of the characteristic-value equation for the creation operator $a\ifmmode\dagger\else\textdagger\fi{}$ is examined. Various proofs are given to demonstrate that the characteristic kets of $a\ifmmode\dagger\else\textdagger\fi{}$ are not physically admissible states of the radiation field. The possible existence of other useful basis states besides ∣n〉|n〉∣n〉, $|\ensuremath{\alpha}〉$, and states generated from these by unitary transformations is then considered. It is shown that when certain restrictions are placed on the correspondence between Hermitian combinations of the arbitrary non-normal operators bbb and $b\ifmmode\dagger\else\textdagger\fi{}$ and the harmonic-oscillator variables xxx, ppp, and HHH, then the only possible basis states are the coherent states $|\ensuremath{\alpha}〉$ and the number states ∣n〉|n〉∣n〉. A $\ensuremath{\lambda}$-dependent variation on the photon annihilation operator aaa is also considered. Its characteristic states for $\ensuremath{-}1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1$ are derived, and shown to form a complete set.
Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.