User:Ezadshojaee/draft article on QND

Quantum non-demolition measurement (QND) is a type of measurement that precisely reproduces the properties of the abstract quantum measurement determined by von Neumann’s postulate of reduction. Quantum non-demolition measurement of an observable is defined as "a sequence of precise measurements of it such that the result of each measurement (after the first one) is completely predictable from the result of the preceding one". The observable which can be measured in this way is called "quantum non-demolition observable". The evolution of a QND observable calculated using the equations of motion in the Heisenberg picture, is unaffected by interaction with the measuring apparatus. Thus, the expectation value and variance of the QND observable evolve during a measurement exactly like when the measuring apparatus has been disconnected, i.e. the "noise" in the measuring apparatus does not feed back on to the observable to increase its variance.

History
The problem of quantum non-demolition (QND) measurement goes back to 1930's when the question of whether an observable of a system can be measured with arbitrary high precision or not was brought up. Before 1960's the interest in quantum measurements with single objects did not grew that much because physicists used to deal with serial tests where desirable precision can be achieved by an increase in the number of tests. With the emergence of quantum electronics and nonlinear optics, QND did become of interest and the mathematical machinery for the measurement theory developed. The investigation of quantum non-demolition measurement was stimulated by the wish to monitor a classical force acting on a harmonic oscillator with better accuracy than can be obtained using standard techniques, which led to the quantum non-demolition measurement proposal in 1974.

Definition
If we do a precise measurement on a system (by coupling it strong enough to a measuring system) we localize it at the measured value of its observable at that time. This initial precise measurement can be regarded as preparing the system in a state with a nearly definite value of the measured observable. The goal of the subsequent measurement is to determine this value. However, the initial, precise measurement inevitably produces uncertainties in observables that do not commute with the measured observable, and in general, these uncertainties "feedback" into the measured observable as the system evolves. Consequently, the result of the subsequent measurement is uncertain. If one wishes to make repeated precise measurements whose results are completely predictable, one must measure an observable that does not become contaminated in other, non-commuting observables. Thus, the system being measured must be in an eigenstate of the measured observable at the time of each measurement. Then the result of each measurement is equal to the eigenvalue at the time of the measurement, and immediately after the measurement the system is left in the same eigenstate, which is left unchanged by each measurement except an unknown phase factor. Therefore, quantum non-demolition measurement of an observable is defined as "a sequence of precise measurements of it such that the result of each measurement (after the first one) is completely predictable from the result of the preceding one". The observable which can be measured in this way is called "quantum non-demolition observable". Thus, "repeatability" is the key feature of a QND.

To formulate the above ideas, consider a sequence of measurements of A in the Heisenberg picture. The initial measurement is made at time t0, where the normalized eigenstates of A(t0) are denoted by |A,α>, where A(t0)|A,α> = A |A,α>, Where α labels the states in any degenerate subspaces of A(t0). The result of the initial measurement is one of the eigenvalues A(t0) of A(t0), and the state of the system immediately after the measurement is an eigenstate of A(t0): |Ψ(t0)=∑αcα|A0,α> with eigenvalue A0 where the cα's are arbitrary (subject to normalization) constants. During the free evolution before the next measurement, the state of the system does not change in the Heisenberg picture. So, if a second measurement is to yield a completely predictable result, then all of the states |A0,α> must be eigenstates of A(t1) with the same eigenvalue, although the new eigenvalue need not equal A0. Hence, one obtains the requirement

A(t1)|A0,α>=f1(A0)|A0,α> for all α

where f1 is an arbitrary real-valued function. It guarantees that the result of a measurement at t=t1 will be f1(A0), because |Ψ(t0> will be an eigenstate of A(t1) with eigenvalue f1(A0) for arbitrary cα's. By assumption, the result of the initial measurement can be any of the eigenvalues of A(t0). Thus, The equation above must hold for all values of A0, and A(t1) must satisfy the operator equation A(t1)=f1[A(t0)]. In a sequence of measurements a similar operator equation must hold at each step in the sequence. Therefore, one obtains the following set of requirements for a QND observable that is to be measured at times t=t0,...,tn:

A(tk)=fk[A(t0)]  for k = 1,...,n

These constraints on the free evolution of A in the Heisenberg picture embody the fundamental principle of QND measurement: If the system begins in an eigenstate of A, its free evolution must leave it in an eigenstate of A at the time of each measurement. That's why a QND measurement precisely reproduces the properties of the abstract quantum measurement determined by von Neumann’s postulate of reduction. It gives - as a result - one of the eigenvalues of the measured observable and projects the measured system on the corresponding eigenstate. Provided it is also an eigenstate of the free Hamiltonian, there is no evolution after the measurement. Therefore, subsequent QND measurements give the same result again and again.

So, the important commutation relation satisfied by any continuous QND observable is

[A(t1),A(t2)]=0  for all times t1 and t2.

One can show that a QND observable can be isolated from the measuring apparatus :

AH(t)=AI(t) Where AH(t) and AI(t) are in Heisenberg and interaction pictures, respectively. The meaning of this fundamental property is that the evolution of a QND observable, calculated using the equations of motion in the Heisenberg picture, is unaffected by interaction with the measuring apparatus. Thus, the expectation value and variance of A evolve during a measurement exactly like when the measuring apparatus has been disconnected. And therefore, the "noise" in the measuring apparatus does not feed back onto A to increase its variance.

QND measurement of small photon numbers
A seminal work which demonstrates QND measurement is the QND measurement of small photon numbers. The probe is a beam of atoms excited with laser into a Rydberg level (levels with high principal quantum number e.g. n=50) before crossing the cavity sustaining the field. The slow spatial variation of the field inside the cavity is essential to decrease the absorption probability. The Hilbert space constitutes of three Rydberg levels f, e , and i (see the figure). The detuning between the cavity mode and the e-i transition is large enough to preclude photon absorption. Yet, the highly polarizable level e experiences a sensible dynamical Stark shift in a single-photon field. The cavity is placed between two field zones R1 and R2 which drive the f-e transition (Ramsey separated-oscillatory-field method). This transition is detected after R2 zone by an atomic ionization counter which discriminates the states e and f. One can show that the presence of N photons in the cavity results in a phase shift, proportional to N of the e state amplitude relative to f which alters the probability of detecting the atom in e or f: Δ(r,N)=[E2(r)d2/ħ2δ)N

Where E(r) is the rms vacuum field in the cavity,d is the electric dipole matrix element for e transition i (δ is shown in the figure) and N is the number of photons in the cavity. The effect of the R1 and R2 oscillatory fields is that the fringe pattern allows one to distinguish a coherent from a thermal or a Fock-state field. . Therefore, monitoring the transfer rate does act as a measurement on the number of photons without absorbing them.



Role of QND in Quantum Control
In feedback loops of control tasks, the controller compares the signal measured by a sensor (output) with the target value and then adjusts an actuator (input) to stabilize the signal around the target value. Generalizing this scheme to stabilize a quantum state of a microsystem relies on quantum feedback, which must overcome a fundamental difficulty: "The sensor measurements cause a random back-action on the system". Sayrin et al. implemented a real-time stabilizing quantum feedback scheme that prepares on demand photon number states (Fock states) of a microwave field in a superconducting cavity, and subsequently reverses the effects of decoherence-induced field quantum jumps. The sensor is a beam of atoms crossing the cavity, which repeatedly performs weak quantum non-demolition measurements of the photon number. The controller is implemented in a real-time computer commanding the actuator, which injects adjusted small classical fields into the cavity between measurements. The microwave field is a quantum oscillator usable as a quantum memory or as a quantum bus swapping information between the atoms.