Quantum nondemolition measurements

ρ(p) = 1

Prob(p)hp|ρ⁰_OP|pi. (1.40)

It is now possible to express the initial state of the probe in a spectral decomposition, with the probabilityw_j associated to the eigenstates|ψ_ji. The action of the measurement on the state of quantum object, conditional to the outcome p, can be expressed as superoperator in a Kraus representation

˜

ρ(p) = 1 Prob(p)

X

j

w_jΩ_jpρ_OΩ^†_jp, (1.41) where the operators Ω_jp defined by

Ωjp =hp|U|ψji. (1.42)

It is straightforward to see that they satisfy the relation P

jΩ_jpΩ^†_jp=I. From this deriva-tion it becomes clear how a nonorthogonal measurement arises as a restricderiva-tion to the Hilbert space of the quantum object of an orthogonal measurement performed on the larger Hilbert space of the probe and the object.

Indirect measurements play a fundamental role in the study of quantum measurement, as they can accurately describe strong measurement as well as weak measurement. Further-more, besides taking into account a detailed microscopic model for the interaction between the object and the probe, indirect measurements are suitable for taking into account the interaction with the environment and achieve a deeper understanding of the measurement process.

1.2.3 Quantum nondemolition measurements

The Heisenberg uncertainty relation is one of the cornerstones of the theory of quantum mechanics. It states that for every quantum object any pair of conjugate variables cannot have a simultaneous precise values. It also represents a fundamental property of a mea-surement process, according to which it is not possible to obtain a simultaneous arbitrarily high precise knowledge of two conjugate variables. Consider the positionxand momentum p of a free particle of massm. The Heisenberg uncertainty relation states that

∆x∆p≥ ~

2. (1.43)

This means that the higher the precision in the determination of, say, the momentum p, and consequently the smaller ∆p, the larger ∆x needs to be. Which is now the limit that quantum mechanics allows for a precise measurement? In order to answer this question we consider two measurements of the position of a free particle of mass m [22, 23]. The first measurement is characterized by an error ∆x₁ in the value of the position x. After a

16 Chapter 1. Introduction

timeτ we perform a second measurement of the positionx, that would yield an error ∆x₂. Between the two measurements, the spread in the momentum due to the error in the first measurement generates a further error δx in the position given by

δx= ∆p₁τ /m=~τ /2m∆x₁. (1.44) From the result x₁ and x₂ of the two position measurements we can infer the value of the momentum p as

p=mx₁−x₂

τ , (1.45)

that will be affected by a total error given my the root mean square (rms)

∆p= m τ

q

∆x²₁+ ∆x²₂+δx². (1.46) By minimization of this expression we find that the optimal solution to obtain that most precise measurement of the momentum is to choose ∆x₁ =p

~τ /2m, from which follows

∆p≥∆p_SQL = r

~m

2τ , (1.47)

and analogously follows that ∆x_SQL =p

2τ /~m. These represent the standard quantum limit (SQL) for a quantum measurement [22, 23].

A natural question is whether it is possible or not to overcome the standard quantum limit. Let us consider a measurement of the energy of an electromagnetic resonator. Ac-cording to quantization of the harmonic oscillator, the energy comes in discrete quanta of energy E =~ω, withω the frequency of oscillation of the resonator, and it is proportional to the number n of excited quanta that is in turn accessible by sending the resonator sig-nal through an amplifier and by measuring the amplitude of the oscillation. Along with the amplitude, the phase φ of the oscillation can be extracted from the amplifier’s output [22]. The energy and the phase are canonically conjugate variables and the Heisenberg uncertainty relation holds

∆E∆φ≥ ~ω

2 . (1.48)

In order to overcome the standard quantum limit in the measurement of the energy of the resonator, a measuring device should respond only to energy and should not acquire any information about the phase. An example of such a device is a photon counter, but such a device performs a strong direct measurement of the oscillator, ultimately absorbing all its energy. We now couple the resonator, whose energy we want to measure, to another resonator that acts as a quantum probe, and perform a strong direct measurement of the photon number of the probe resonator with a photon counter. If the energy of the first resonator does not change during the measurement of the probe we can perform a measurement that “deceives” the quantum limit. Such a measurement goes under the name of quantum nondemolition measurement (QND).

1.2 The measurement process 17

As a general recipe, we consider a quantum system on which we want to measure a suitable observable ˆA. A measurement procedure is based on coupling the system under consideration to a probe. The global evolution entangles the probe and the system, and a measurement of an observable ˆB of the probe provides information on the system. In general, a strong projective measurement on the probe translates into a weak non-projective measurement on the system (see Sec. 1.2.2). This is because the eigenstates of the coupled system differ in general from the product of the eigenstates of the measured observable on the system and those of the probe.

Three criteria that a measurement should satisfy in order to be QND are [24]: i) agree-ment between the input state and the measureagree-ment result; ii) the action of measureagree-ment should not alter the observable being measured; iii) repeated measurement should give the same result. These three criteria can be cast in a more precise way: the measured observable Aˆmust be an integral of motion for the coupled probe and system[22]. Formally this means that the observable ˆA that we want to measure must commute with the Hamiltonian H, that describes the interacting system and probe,

[H,A] = 0.ˆ (1.49)

Such a requirement represents a sufficient condition such that an eigenstate of the ob-servable ˆA, determined by the measurement, does not change under the global evolution of the coupled system and probe. As a consequence, a subsequent measurement of the same observable ˆA provides the same outcome as the previous one with certainty. As a counterexample for a case in which a quantum nondemolition measurement is not possible, one can consider the measurement of the position of a free particle, for which the system observable is ˆA = ˆx and the Hamiltonian of the system is H = ˆp²/2m. Clearly one has [H,x]ˆ 6= 0 and a QND scheme does not work.

Finally, in order to obtain information on the system observable ˆAby the measurement of the probe observable ˆB, it is necessary that the interaction Hamiltonian doesnotcommute with ˆB,

[H_int,B]ˆ 6= 0, (1.50) where Hint describes the interaction between the probe and the system,

H =H_S +H_probe +H_int. (1.51)

Altogether, these criteria provide an immediate way to determine whether a given mea-surement protocol can give rise to a QND meamea-surement.

Conclusion

The ideas of quantum measurement and the formalism introduced in this section find an application in Ch. 3 where we study the quantum nondemolition measurement of a qubit coupled to a harmonic oscillator. In particular, the concept of indirect measurement, with a system represented by a qubit and a probe represented by a harmonic oscillator, and the POVM formalism for weak measurements provide us with the mathematical tools that

18 Chapter 1. Introduction

allows us to model a sequence of two qubit measurements and to study the correlations in the conditional probability for the measurement outcomes.