rydberg-atom phase sensitive detection - uq espace157843/n07chapter6.pdfrydberg-atom phase sensitive...

Chapter 6

Rydberg-atom phase sensitive

detection

In the previous chapters of this thesis we have considered first, quantum measurement theory, and

then, the measurement interpretation of a recent test of the quantum Zeno effect. In the remainder of

this thesis, we examine several proposals for either tests of the quantum Zeno effect, or for proposed

measurement interactions that could be employed in tests of the quantum Zeno effect.

In this chapter we consider a recently proposed phase sensitive detection scheme and examine its

application to the generation of the quantum Zeno effect. Throughout, we emphasise the need to analyse

the measurement interaction in all its stages, and to distinguish adequately between measurement caused

detuning effects, and the actual measurement of the dynamics of the system of interest.

The system of interest consists of a single two-level Rydberg atom exchanging one photon with a

single cavity mode of the electromagnetic field. The photon number occupancy of the cavity mode is then

monitored using a beam of Rydberg atoms configured so as to perform phase sensitive detection. Because

the detector consists of a beam of atoms passing through the cavity, the measurement interaction consists

of a discrete series of pulsed measurements. The continuous measurement limit can be approached by

increasing the spatial density of the monitoring Rydberg atom beam. In this limit we derive a master

equation for the evolution of the cavity mode when the photon number is monitored by this method. This

equation describes a phase diffusion process. In the limit of very rapid monitoring, the free oscillation

of the atomic inversion is disrupted and the atom can be trapped close to the initial excited state. This

is the quantum Zeno effect.

91

92 CHAPTER 6. RYDBERG-ATOM PHASE SENSITIVE DETECTION

6.1 Introduction

Recently a very interesting photon number quantum nondemolition measurement was proposed by Brune

et al. [82, 83], based on the interaction between a Rydberg-atom transition detuned from a microwave

cavity. This scheme opens up a more direct test of the Zeno effect in a two level system with dynamics

dominated by coherent oscillation (i.e. there is no need to consider spontaneous emission). The general

scheme is as follows. A single two-level Rydberg-atom is placed in a high Q microwave cavity resonant

with the atomic transition. If the atom is prepared initially in the excited state the evolution of the

atom-cavity system will consist in a coherent oscillation between two states corresponding to the atom

initially excited and no photons in the field, and the atom in the ground state and one photon in the

field. The initial state is monitored by a quantum nondemolition (QND) measurement of the photon

number in the cavity. The QND scheme is the Rydberg-atom phase sensitive detection scheme proposed

by Brune et al. As we shall show when the measurement is operating to extract the maximum amount

of information at the greatest rate the coherent oscillation in the two state system is suppressed and the

evolution is dominated by a very slow decay of the initial state.

6.2 Photon number QND measurements using Rydberg-atoms

We now describe in some detail a photon number QND scheme essentially equivalent to the scheme of

Brune et.al. [82, 83]. As depicted in Fig. 6.1, a beam of Rydberg atoms passes into a microwave cavity

of resonant frequency ωc. The level structure of the atoms is also indicated in Fig. 6.1. The |2〉 ↔ |3〉transition is coupled to the cavity field but is widely detuned. This ensures that there is no absorption

of photons from the cavity due to this transition. Prior to entering the cavity the Rydberg-atoms pass

through an intense field L1 resonant with the |1〉 ↔ |2〉 transition. This field is to prepare the atoms in a

superposition of states |1〉 and |2〉. After leaving the cavity the atoms pass through a similar intense field

L2 which is π/2 out of phase with the first field, and then into an ionization counter which determines

whether the atom is in state |2〉. In the scheme of Brune et al. the fields L1, L2 form a Ramsey fringe

experiment, however we will not view it in quite this way.

The interaction with the fields L1, L2 may be described in terms of a rotation of the Bloch vector

representing the inversion and polarization of the |1〉 ↔ |2〉 transition. For the system in this chapter

the inversion and polarization operators are

Jz =12

(|2〉〈2| − |1〉〈1|)

Jy = − i

2(|2〉〈1| − |1〉〈2|)

Jx =12

(|2〉〈1| + |1〉〈2|) . (6.1)

Operators equivalent to these were introduced in Chap. (4) for an arbitrary two level system. The

6.2. PHOTON NUMBER QND MEASUREMENTS USING RYDBERG-ATOMS 93

L1 L2

cavity

atomicbeam

ionisationcounter

1

2

3

ωL

ωc

Schematic representation of the QND scheme to measure the photon number in the cavity.

L1 is a field used to prepare the state of the atoms so that they have a non-zero dipole on

entering the cavity, while L2 ensures that the final ionization count will give information on the

photon number in the cavity.

Figure 6.1: The schematic experimental setup

interaction with the fields L1, L2 are then described by the unitary operators

R1(φ1) = e−iφ1Jx

R2(φ2) = e−iφ2Jy . (6.2)

The phase of precession, φj , is proportional to the product of the dipole moment for this transition, the

interaction time and the total field strength. We will assume that all atoms entering the field L1 are in

the ground state |1〉. Thus after passing through the first field the state of an atom is

|ψA〉 = cos(φ1/2)|1〉 − i sin(φ1/2)|2〉. (6.3)

This first field may be regarded as a state preparation step for the probe atoms.

Inside the cavity sufficiently far from resonance, the interaction is described by the Hamiltonian [85]

HI = hχσ23z a†a (6.4)

where

σ23z =

12

(|3〉〈3| − |2〉〈2|) , (6.5)


and a†a is the photon number operator for the intracavity field. Clearly a†a is a QND variable for the

cavity field, and HI represents a back-action-evasion (BAE) coupling.

To see how the QND measurement works consider the Heisenberg equations of motion for the dipole

moment operators on the |1〉 ↔ |2〉 transition, Eq. (6.1),

dJx

dt=

χ

2Jya†a

dJy

dt= −χ

2Jxa†a. (6.6)

As a†a is a constant of motion we obtain the solutions for an interaction time τ as

Jx(τ) = cos(θa†a)Jx(0) + sin(θa†a)Jy(0)

Jy(τ) = cos(θa†a)Jy(0) − sin(θa†a)Jx(0), (6.7)

where θ = χτ/2. Thus a measurement of Jx or Jy will yield information on a†a, provided the state of

the atom is such that there is a non-zero dipole on the transition |1〉 ↔ |2〉, (i.e. 〈Jx〉, 〈Jy〉 6= 0 ). The

purpose of the initial field is to ensure that this is the case. Unfortunately the ionization counter at the

output of the experiment effectively measures Jz not the dipole moment operators. The purpose of the

second field is to rotate the information in the dipole into a component of the inversion.

If we view the experiment as a whole (Fig. 6.1), it effectively transforms the input Bloch vector

components of the |1〉 ↔ |2〉 transition into output components. The final measurement is made on the

z component of the output. The total transformation of the z component is

Joz =

(cos φ1 cos φ2 + sinφ1 sin φ2 sin(θa†a)

)J i

z − sinφ2 cos(θa†a)J ix

+(sin φ1 cos φ2 − sin φ2 cos φ1 sin(θa†a)

)J i

y. (6.8)

Note that if the atom is not first prepared in a superposition state by the first laser field, φ1 = 0 and

〈J iy〉 = 〈J i

x〉 = 0 so no information on a†a is obtained, i.e. no measurement has taken place. If we now

take φ1 = φ2 = π/2 ,

Joz = sin(θa†a)J i

z − cos(θa†a)J ix. (6.9)

Effectively with this choice of phase the transformation is a precession about the y-axis of the Bloch

sphere, a result easily confirmed by a geometric representation of each rotation.

The mean signal at the detector is

〈Joz 〉 = −1

2〈sin(θa†a)〉. (6.10)

If θ is small and only low photon numbers are excited in the cavity,

〈Joz 〉 ≈ −θ

2〈a†a〉. (6.11)

This is clearly a measurement of the photon number in the cavity.

In the case of θ large, a measurement of Jz will condition the cavity mode into a superposition of states,

each of which individually has photon number occupancy satisfying Eq. (6.10). Similar measurement

generated superposition states are considered in more detail in Chap. (8).

6.3. INTRACAVITY DYNAMICS 95

6.3 Intracavity dynamics

We now determine the evolution equation for the measured cavity field state and show that the mea-

surement leads to a rapid diagonalization of the state of the field in the number basis, that is the state

is reduced as a result of the measurement. The state of each atom entering the cavity is the state after

the field L1.

|ψA〉 = c1|1〉 + c2|2〉, (6.12)

where c1 = cos(φ1/2), c2 = −i sin(φ1/2). The change in the state of the field due to the interaction of a

single atom for a time τ is

ρ′ = Φτ ρ

≡ trA

(U(τ)ρ⊗|ψA〉〈ψA|U†(τ)

)= |c1|2ρ + |c2|2eiθa†aρe−iθa†a. (6.13)

In this equation U(τ) is the time evolution operator following from the interaction Hamiltonian in

Eq. (6.4).

The transformation defined by Eq. (6.13) is nonunitary. The resulting state is a statistical mixture of

a field which has undergone a phase jump of θ and a state which is left unchanged. The probability that

a given probe atom will induce a phase change is |c2|2. Note that the phase jump, if it occurs, is always

of the same size and direction. The nonunitarity arises because we do not know whether a given probe

atom has induced a phase jump or not.

We first consider the rather idealised case where θ does not vary from atom to atom and all the atoms

arrive in the cavity at equally spaced time intervals. In this case Eq. (6.13) can be iterated N times to

give

ρ(N) =N∑

r=0

(N

r

)|c2|2r|c1|2(N−r)eirθa†aρ(0)e−irθa†a. (6.14)

(We ignore for the moment the free evolution between measurements). To understand the effect of the

measurements on the field consider the matrix elements of ρ(N) in the number basis

ρ(N)p,q = 〈p|ρ(N)|q〉

= ρ(0)p,qe

i θ2 (p−q)N

(|c1|2e−i θ

2 (p−q) + |c2|2ei θ2 (p−q)

)N

. (6.15)

In the case |c1| = |c2| = 1/√

2 we find

ρ(N)p,q = ρ(0)

p,qei θ2 (p−q)N

[cos

(θ

2(p − q)

)]N

. (6.16)

We now consider a continuous regular limit in which θ is small but N is large. (That is each measurement

is not very effective but we make very many to compensate). To second order in θ we find

ρ(N)p,q ≈ ρ(0)

p,qei θ2 (p−q)N exp

(−Nθ2

8(p − q)2

). (6.17)


Clearly there is a decay of coherence in the number basis, a result to be expected for a scheme designed

to measure the number. If we assume that the time between kicks is T then N = t/T and

ρ(N)p,q ≈ ρ(0)

p,qei θ2 (p−q)γt exp

(−γtθ2

8(p − q)2

). (6.18)

where γ = T−1 is the frequency of the measurements; effectively the bandwidth of the measurement.

In this form the decay of coherence is typical of the decay of coherence in continuous measurement

models [3]. Clearly a good measurement corresponds to γθ2 large. The issue of whether measurement

generated detuning effects dominate in this regime will be addressed below.

Away from the continuous limit coherence decay is more complicated. In fact for finite θ there are

certain coherences which do not decay. For example if θ = π/2 coherence between states |p〉 and |q〉will not decay whenever p − q = 4n for n an integer. Even so if N is large ρ

(N)p,q decreases very quickly

away from these special values. We will later be interested in interpreting the case θ = π. In this case,

Eq. (6.13) gives (for |c1| = |c2| = 1/√

2),

ρ′ =12

∑n,m

(1 + (−1)(n−m)

)ρnm|n〉〈m|, (6.19)

and we see a very strange pattern of coherence loss with every other matrix element sent to zero.

This strange coherence loss pattern would tend to suggest that setting θ = π does not constitute a

measurement at all. However, we will later show that if we take into account the uncertainty in the

interaciton strength θ, then we do effectively kill off the all the coherences in the photon number basis.

While this is then similar to a measurement interaction, we must complete further analysis before we

can determine whether a measurement has been effected. We will comment further on the interpretation

of this below.

The source of the coherence decay is the need to prepare the initial state of the atom to be a superpo-

sition of the ground and excited states, as noted in the previous section. This is easily seen in Eq. (6.15)

by setting either c1 or c2 to zero. For nonzero c1 and c2 the field then sees a random sequence of excited

and unexcited atoms which leads to a stochastic variation of the refractive index inside the cavity. The

need to prepare the atom in a state suitable for a measurement to take place leads directly to state

reduction in the energy basis. This is a totally unavoidable nonunitary effect of the measurement.

However other sources of nonunitarity may also play a role. For example the atoms may arrive in the

cavity at random times rather than at regular time intervals as assumed above. In the case of Poisson

distributed arrival times an evolution equation may be derived for the cavity field [51],(dρ

dt

)meas.

= γ(Φτ ρ − ρ)

= γ|c2|2(eiθa†aρe−iθa†a − ρ

). (6.20)

In addition the interaction time τ may vary from atom to atom due to the velocity profile. In this case

we must average over a distribution for θ. Assuming a Gaussian distribution of mean θ and a variance

6.3. INTRACAVITY DYNAMICS 97

of ∆, such that θ À √∆ then to first order in the variance(

dρ

dt

)meas

= γ|c2|2(eiθa†aρe−iθa†a − ρ

)− Γ∆

[a†a, [a†a, eiθa†aρe−iθa†a]

], (6.21)

where

Γ∆ =∆γ|c2|2

2. (6.22)

We note that in the photon number basis this reads

˙ρpq =γ

2

(eiθ(p−q) − 1

)ρpq − Γ∆eiθ(p−q)(p − q)2ρpq. (6.23)

This equation makes it obvious that, for ∆ = 0, there always exist certain coherences that do not decay

in this basis. These coherences satisfy θ(p − q) = 2nπ for n an integer. However, the picture is vastly

more complicated in the case where ∆ 6= 0.

An example that will be of later interest is when the cavity mode is restricted to having just zero

or one photon in it. In this case, all coherences but one are identically equal to zero. The equation of

motion for the remaining coherence is

˙ρ01 =γ

2

(eiθ − 1

)ρ01 − Γ∆eiθρpq. (6.24)

Here, we see that, for ∆ = 0 the coherence is always damped unless θ = 2nπ for n an integer, and that

the coherence suffers significant detuning for most values of θ. Two further interesting cases emerge.

Consider the situation where θ = π giving

˙ρ01 = −(γ − Γ∆)ρ01, (6.25)

which represents pure coherence decay for γ À Γ∆. In the case θ = π/2 we have

˙ρ01 =[−γ

2+ i(

γ

2− Γ∆)

]ρ01, (6.26)

giving a damped, and significantly detuned evolution. In our analysis of the measurement interaction,

we will need to take this detuning into account.

We now consider the case where the average phase shift is small, with θ ≈ ∆ ≈ 0, and, if there are

few photons in the cavity we may write(dρ

dt

)meas.

= iδ[a†a, ρ] − Γ[a†a, [a†a, ρ]

], (6.27)

where

δ = γ|c2|2θ, (6.28)

is the linear deterministic phase shift and

Γ = Γ∆ +θγ|c2|2

2(6.29)

is the measurement parameter. The effect of the first term in Eq. (6.27) is to induce a fixed detuning

of the cavity field from the empty cavity frequency. This deterministic phase shift is the average phase


shift induced by a beam of probe atoms which enter the cavity at random, Poisson distributed times.

The second term leads to a diffusion in the phase of the cavity field. We note that any uncertainty in

the velocity profile (∆ > 0), of necessity, leads to enhanced phase diffusion. More importantly this last

term also causes the density operator for the field to become diagonal in the photon number basis, a

result to be expected for a measurement of the photon number. A good measurement thus corresponds

to large Γ, as in the regularly measured case. Again, as with the Itano experiment of Chap. (5) we

must take careful account of the measurement induced detuning of the system before we can analyse the

consequences on the system dynamics of introducing the measurement interaction.

6.4 Conditional state of the cavity

Given that an atom is detected in state |2〉 what is the state of the cavity field, conditioned on this

result? In this section we give the answer to this question, which is needed to simulate the evolution of

a particular measurement sequence.

The probability to detect an atom in the state |2〉 after passing through the final field, L2 is

P2 = trF

(Φ(2)

τ ρ)

, (6.30)

where

Φ(2)τ ρ = trA

(|2〉〈2|R2(φ2)U(τ)ρ ⊗ ρAU†(τ)R†

2(φ2))

= 〈2|R2(φ2)U(τ)ρ ⊗ ρAU†(τ)R†2(φ2)|2〉, (6.31)

and where ρA is the state of the atom after passing through L1 [given in Eq. (6.12)]. The state trans-

formations in the final rotation of the Bloch vector are

R2(φ2)|1〉 = d1|1〉 − d2|2〉R2(φ2)|2〉 = d2|1〉 + d1|2〉, (6.32)

where d1 = cos(φ2/2) and d2 = sin(φ2/2). Using Eqs. (6.12,6.31) and (6.32) we find

Φ(2)τ ρ = |c1|2|d1|2ρ + |c2|2|d2|2eiθa†aρe−iθa†a − c1c

∗2d2d

∗1ρe−iθa†a − c∗1c2d

∗1d2e

iθa†aρ. (6.33)

Thus

P2 = |c1|2|d1|2 + |c2|2|d2|2 −(c1c

∗2d2d

∗1〈e−iθa†a〉 + c.c

), (6.34)

where c.c. denotes the complex conjugate. Note that if either c1 or c2 is zero, no information on the

cavity photon number is obtained. In the case of φ1 = φ2 = π/2 the coefficients are c1 = 1/√

2, c2 =

−i/√

2, d1 = d2 = 1/√

2 and

Φ(2)τ ρ =

14(ρ + eiθa†aρe−iθa†a − iρe−iθa†a + ieiθa†aρ), (6.35)

6.4. CONDITIONAL STATE OF THE CAVITY 99

and thus

P2 =12− 1

2〈sin θa†a〉. (6.36)

As 〈Joz 〉 = P2 − 1

2 this result agrees with the result in Eq. (6.10). If the field is in a coherent state |α〉we find

〈sin θa†a〉 = exp(−|α|2(1 − cos θ)

)sin(|α|2 sin θ), (6.37)

which for θ ¿ 1 becomes approximately sin(|α|2θ), the semiclassical result.

The conditional state of the field is then given by

ρ(2) = (P2)−1 Φ(2)

τ ρ

=ρ − iρe−iθa†a + ieiθa†aρ + eiθa†aρe−iθa†a

2(1 − 〈sin θa†a〉) . (6.38)

The reduced photon number distribution in particular is

P (2)(n) =P (n)(1 − sin θn)1 − 〈sin θa†a〉 . (6.39)

This will have holes at values of n such that nθ = π(2m+ 12 ) for m an integer. In a long sequence of mea-

surements the nett effect of superposing different interference patterns corresponding to the stochastic

sequence of results is that the reduced probability distribution converges to a single peak at a random

value of n, as demonstrated in Brune et al. [82, 83]. In fact for certain values of θ, conditional measure-

ments can result in a photon number distribution that is multiply peaked. This would require a beam

for which the standard deviation of θ is much less than the mean. Atomic beams can be prepared with

a velocity distribution width of 4% [87]. For completeness, the operation for the conditional state of the

field given that the atom was not detected in state |2〉 is

Φ(1)τ ρ =

14(ρ + eiθa†aρe−iθa†a + iρe−iθa†a − ieiθa†aρ). (6.40)

Thus

P1 =12

+12〈sin θa†a〉. (6.41)

If θ = π/2 and there is one photon in the field we find that P2 = 0 and P1 = 1. That is, the atom

will never be ionised from state |2〉, but only from state |1〉 in the final ionization state readout. It then

seems that the best way to detect a single photon would be to arrange to have θ = π/2. This is of

relevance to the quantum Zeno effect determination discussed in the next section.

Although in a particular sequence of measurements a random time distribution of ionization counts

will be observed the average ionization rates are easily calculated. The average ionization rate from

states |1〉, or |2〉 is given by

i1,2(t) = γP1,2

=γ

2(1 ± 〈sin θa†a〉), (6.42)


where γ is the atom injection rate. We will employ this result in analysing the best measurement regimes

in the next section. Here we see that the ionization rate depends both on θ and the average photon

number occupancy of the field, as required. For photon number equal to one, the best measurement

regime will have θ = π/2. We confirm this below.

We now take into account the uncertainty in the interaction strength as before and consider θ to have

mean θ and variance ∆. As well we consider the case where the cavity mode has just zero or one photons

in it, with respective probabilities ρ00 and ρ11. Then, to first order in the variance, the average ionization

currents are

i1,2(t) =γ

2(1 ± ρ11 sin(θ). (6.43)

It is immediately apparent that the random variation in the average interaction will not effect the

average ionization current. Further, we note that the case θ = π does not provide any mechanism

for information flow from the microscopic to the macroscopic realm. This case does not satisfy the

minimum requirements for a measurement to occur. There is a significant coherence loss mechanism,

but this mechanism does not constitute a measurement of the system of interest. Thus, we have a similar

situation to that considered in Chap. (5). The case θ is akin to collisional damping, and will be discussed

in more detail later in this chapter.

As mentioned above, the case θ = π/2 does provide a channel for information flow about the system to

the experimenter. This might then be the good measurement regime. However, as shown in Eq. (6.26),

this case also generates significant detuning of the system of interest, and so we must consider the

magnitude of this detuning before we can say whether we are effecting a measurement. This is also

similar to the approach considered in previous chapters.

For completeness we also give the state of the field conditioned on the occurrence of an interaction

but where no account is taken of the results of that interaction. This is

Φτ ρ = Φ1τ ρ + Φ2

τ ρ

=12

(ρ + eiθa†aρe−iθa†a

). (6.44)

This equation is equivalent to Eq. (6.13) and demonstrates the expected coherence loss mechanisms.

This equation is later expanded in the Poisson injection case in Eq. (6.65).

6.5 A test of the quantum Zeno effect

We now assume that the microwave cavity initially contains a single two-level Rydberg-atom, referred

to as the object atom, with a transition frequency resonant with the cavity frequency. The excited state

of this atom will be denoted |e〉 while the ground state is |g〉. We assume also that the atom is prepared

initially in the excited state and that the cavity contains no photons. Under free evolution the quantum

of energy is periodically exchanged between the object atom and the cavity field. By monitoring the

6.5. A TEST OF THE QUANTUM ZENO EFFECT 101

photon number in the cavity we can monitor the evolution of the atom away from its initial excited

state. We will discuss separately the two cases of regular probe injection and Poisson probe injection.

We show in both cases that the measurement necessarily disrupts the free oscillation of the atom/field

system.

6.5.1 Regular probe injection

In Sect. (6.3) we showed that the interaction necessarily introduces a minimum degree of nonunitary

change in the field state, described by Eqs. (6.13) or (6.44). This change is interpreted as a statistical

mixture of a field which has undergone a phase change and a field which has not. In the case |c1|2 =

|c2|2 = 1/2 there is an equal probability for each of these events. Each probe atom has a 50% chance

of causing a fixed phase change in the state of the field. In this section we show that this minimal level

of nonunitarity induced by the measurement causes a change in the free evolution of the cavity/object

atom system which becomes more disruptive the greater the rate of measurement.

Under free evolution the object atom will periodically emit and absorb one photon; an entirely coherent

process in the absence of spontaneous emission (we assume that the rate of spontaneous emission for

this configuration is much smaller than the coherent oscillation frequency). The Hamiltonian for this

interaction is

HI =hκ

2(a†σ− + aσ+), (6.45)

where σ± are the dipole raising and lowering operators for the |e〉 ↔ |g〉 transition. The initial state is

|e〉⊗ |0〉F i.e. the atom is in the excited state and the field is in the vacuum state. The state at any time

lies within the subspace spanned by

|a〉 = |e〉 ⊗ |0〉F|b〉 = |g〉 ⊗ |1〉F . (6.46)

In this basis the matrix elements of the density operator obey

d~S

dt= ~B × ~S, (6.47)

where

~S = (X,Y, Z)

~B = (κ, 0, 0), (6.48)

and for convenience we rewrite the components of the Bloch vector for this system as

X =12(〈a|ρ|b〉 + 〈b|ρ|a〉)

Y =i

2(〈b|ρ|a〉 − 〈a|ρ|b〉)

Z =12(〈b|ρ|b〉 − 〈a|ρ|a〉). (6.49)


These equations represent the precession of ~S around the direction defined by ~B, ie the x−axis.

We will assume that the interaction of the probe atom with the cavity field occurs on a much faster

time scale than the coherent interaction between the cavity field and the object atom. This requires that

the probe atoms pass through the cavity at a very high rate. This point is further discussed later in this

chapter. With this assumption the effect of the probe atom on the system may be modelled as a “kick”,

which periodically interrupts the free dynamics. The dynamics is then represented by a map for the

vector ~S which comprises a free precession for a fixed time followed by a nonunitary jump as the probe

atom passes through the cavity. To determine this map we first note that the interaction Hamiltonian

may be written as

HI = hκσx, (6.50)

where the angular momentum operators are defined with respect to the effective two level system by

σx =12(|b〉〈a| + |a〉〈b|)

σy =−i

2(|b〉〈a| − |a〉〈b|)

σz =12(|b〉〈b| − |a〉〈a|). (6.51)

Thus the free dynamics is simply represented as a precession around the x-axis by an angle φ = κτ

where τ is the time between each probe atom. To determine the effect of a probe atom we note that the

photon number operator in the effective two level system is formally identical to σz,

a†a ⊗ IA = σz, (6.52)

where IA is the identity operator for the object atom. (This result may be verified by checking the

commutation relations). A phase shift of θ in the field is thus represented by a precession of θ about the

z-axis.

The resulting map for the state of the system is

ρn+1 =12

(e−iφσx ρneiφσx + e−iθσze−iφσx ρneiφσxeiθσz

), (6.53)

where θ is the phase change induced by a probe atom and φ = κτ where τ is the period of time between

each probe atom passing through the cavity.

The state ρn+1, is a statistical mixture of states one of which undergoes a precession of φ around

the x−axis and one which undergoes a precession of φ around the x−axis followed by a precession of

θ around the z−axis. Any given probe atom will induce one or the other of these processes with equal

probability. The corresponding map for the vector ~S is

~Sn+1 =12(R1 + R2)~Sn, (6.54)


where

R1 =

1 0 0

0 cos φ − sin φ

0 sinφ cos φ

R2 =

cos θ − sin θ 0

cos φ sin θ cos φ cos θ − sin φ

sin φ sin θ sin φ cos θ cos φ

. (6.55)

In the case of θ small the total map matrix may be written approximately as

R =

cos θ2 − sin θ

2 0

cos φ sin θ2 cos φ cos θ

2 − sin φ

sin φ sin θ2 sinφ cos θ

2 cos φ

×

e−θ2/8 0 0

0 e−θ2/8 0

0 0 1

, (6.56)

which is correct to second order in θ only. In this form we see that the probe atoms on average induce

an extra precession of θ/2 about the z-axis in addition to the unitary precession around the x−axis and

they also induce a nonunitary decay of the “coherences”, Sx and Sy, at the rate θ2/8. This decay is of

course the same as that given in Eq. (6.17). The measurement has both a systematic detuning effect,

as determined by θ/2 and a random or diffusive effect determined by θ2. These results suggest that the

“good measurement” regime, wherein we seek to minimise the magnitude of the detuning compared to

the coherence loss, will lie in the large θ limit.

The probability to find the cavity/object atom system in the initial state after n probe kicks, is given

by

pa,n =12− Sz,n. (6.57)

In Figs. 6.2 and 6.3 we plot this probability versus t = nφ for a number of cases. In Fig. 6.2(a-c)

we consider the case of fixed θ = π/2 (ie fixed measurement strength), but vary the time between

each probe atom, ie γ. This shows the effect of increasing the rate of measurement. It is apparent

that the oscillations in the probability pa are suppressed as the rate of measurement increases, and the

system remains longer near the initial state. The measurement thus disrupts the free evolution as the

measurement rate increases. This is what is meant by the quantum Zeno effect. This figure suggests

that θ = π/2 makes for a good measurement of the system populations. We will establish the “good”

measurement regimes in the next section.

In the explicit model of this chapter we are able to trace the origin of this effect to the physical effect

of the measuring apparatus (the probe atoms) on the measured system (the object atom/field). We

can distinguish two physical explanations for the effect in the model. The first is the average extra

rotation induced by the probe atoms which has the effect of tilting the precession direction away from

the x−axis as described by the first factor in Eq. (6.56). This causes a frequency shift in the evolution

of pa that is apparent in comparing Fig. 6.2(a-c). This “tilting” of the axis of rotation is a detuning of


the system evolution that, as we noted in previous chapters, interferes with any attempts to measure

the undisturbed system evolution.

This is more evident in Fig. 6.3 where we have considered a “weak-coupling” limit in which θ is small

but the product of θ and the measurement rate is greater than one. In this case the effective detuning

dominates the evolution of the survival probability, although a very slow decay is just discernible. This

limit will be discussed in more detail in the next section.

The second explanation is the destruction of coherence, or the decay of polarization, reflected in the

second factor in Eq. (6.56). This decay is due to the fact that a given probe atom may or may not

cause a change in the phase of the field in the cavity, with each event equiprobable. It may of course

be possible, in some experiments, to eliminate the systematic effect of the measurement, perhaps by

arranging for θ to fluctuate between a positive and negative value rather than between a positive value

and zero as in this chapter, but nothing can be done about the fluctuations induced by the measurement.

It is always possible to view the effect of measurement in terms of an ensemble dynamics, as above,

or as a stochastic trajectory conditioned on the history of what each particular probe atom actually

does. In the model of this chapter the actual phase change induced by a probe atom determines the

measured state of the probe atom after it has exited the cavity and is thus known (in principle at least)

to the experimenter. This gives another way to view the dynamics of the vector ~S as the measurement

proceeds by directly simulating the random process represented by the passage of the probe atoms.

In this case we toss a coin to determine whether to apply R1 or R2 at each step of the map. The

results of such a simulation are shown in Fig. 6.4. We know that there is a systematic shift in the

precession axis but this is hard to see in a single trajectory. However, it is now quite apparent that

the measurement causes random phase disturbances to the evolution, much in the way collisions disrupt

the polarization dynamics for atomic transitions. This same measurement generated phase disturbance

was seen in Chap. (5) in Figs. 5.10 through 5.11. There, we showed that atomic level measurements on

a two level system could be described by the measurement operator Jz which caused a phase reversal

of the evolution of the system. The relationship between the proposed measurement interaction of this

chapter, and that discussed previously, is canvassed in the next section.

6.5.2 Poisson distributed probe injection

We now assume that the probe atoms arrive in the cavity at Poisson distributed time intervals. This is

probably a more realistic assumption than the case of regular arrival times discussed above. However

the general picture of the dynamics is not changed much. One considerable advantage of the Poisson

model is that it enables one to write down an explicit evolution equation for the state (or vector ~S),

rather than a map. In this section we the interaction strength of the probe atom in the cavity to be

fixed, that is ∆ = 0.


5 10 15 20 25

0.2

0.4

0.6

0.8

1

p (t)a

t

(a)

5 10 15 20 25

0.2

0.4

0.6

0.8

1 (b)

p (t)a

t

5 10 15 20 25

0.2

0.4

0.6

0.8

1

(c)

p (t)a

t

A plot of the survival probability, pa(t), versus time for a regular injection probe. We choose

θ = π/2 and show the effect of increasing measurement rate in (a) through to (c). The time t

is defined in terms of the number of injections by t = nφ, where n is the number of injected

probe atoms to that point and φ is the time of interaction of the probe atom in units of the

one-photon Rabi period (κ−1). (a) No measurement (b) φ = 1.0 (c) φ = 0.5

Figure 6.2: The initial state survival probability with measurement


5 10

0.2

0.4

0.6

0.8

1

p (t)a

t

The survival probability, pa(t) is plotted versus time for the regular probe injection case.

θ = 0.05 , φ = 0.05 . The effective measurement rate is γ = 20: all in units of κ−1.

Figure 6.3: The initial state survival probability under regular injection

The general form of the evolution equation for a system subjected to Poisson distributed kicks is [15]

dρ(t)dt

=−i

h[H0, ρ(t)] + γ(Kρ(t) − ρ(t)), (6.58)

where K is the operation describing the effect of the kick on the density operator and γ is the average

injection rate. For this approach to be valid we require γ large. In the system of this chapter K = Φτ

is defined by Eq. (6.13). With |c1|2 = |c2|2 = 1/2 the evolution equation with free Hamiltonian given in

Eq. (6.50) becomes

dρ

dt= −iκ[σx, ρ] +

γ

2(eiθσz ρe−iθσz − ρ

). (6.59)

This equation is equivalent to Eq. (6.21) for ∆ = 0 and corresponds to a sequence of Poisson distributed

rotations around the z−axis at half the injection rate γ; not a surprising result as, on average, only half

the atoms are effective in producing a phase change in the field.

Eq. (6.59) gives the resulting dynamics for the vector ~S with time measured in units of κ, as

d

dt

X

Y

Z

=

−β sin2(

θ2

)β2 sin(θ) 0

−β2 sin(θ) −β sin2

(θ2

) −1

0 1 0

X

Y

Z

. (6.60)

where we have defined the dimensionless parameters β = γ/κ.

It is clear from Eq. (6.60) that the measurement has detuned the object atom from resonance with

the cavity. The precession direction is now ~B = (κ, 0, β sin(θ)/2) rather than that specified by the free

dynamics (κ, 0, 0). This detuning is due to the systematic phase shift averaged over a large number of


2 4 6 8 10

0.2

0.4

0.6

0.8

1

p (t)a

t

A stochastic simulation of the survival probability pa(t) when account is taken of each probe

atom. At each step there is a probability of 0.5 that a probe atom will cause a phase shift in

the field, or do nothing. θ = 0.3 , φ = 0.2. The effective measurement rate is γ = 5: all in

units of κ−1.

Figure 6.4: The initial state survival probability under selective measurement interactions

probe atoms. Also apparent is the measurement generated coherence loss mechanisms proportional to

β sin2(θ/2).

Small θ limit

The dynamics described by Eq. (6.59) can be illustrated in the small θ regime where we can expand the

master equation [Eq. (6.59)] as

˙ρ = −iκ[σx, ρ] − iδ[σz, ρ] − Γ [σz, [σz, ρ]] , (6.61)

where in this section we assume the atom-cavity interaction time is constant (∆ = 0, θ = θ) and so from

Eq. (6.29) we have

Γ =γθ2

4

δ =γθ

2. (6.62)

From this, we readily see that, for small θ the measured system is subject to a measurement detuning

proportional to δ and a measurement caused rate of coherence loss proportional to Γ. Here, we see clearly

the relationship between this proposed measurement interaction and that described by Jz as mentioned

briefly in the previous section. [See Chap. (5) and Figs. 5.10 through 5.11]. Again we see the master

equation reducing to the form of the Angular Momentum model introduced in Chap. (3).


This equation reinforces one of the conclusions of Chap (5), namely that any interpretation of a

measurement interaction must take explicit account of any measurement induced detuning of the system

dynamics as well as examining mechanisms for the measurement caused loss of coherence. In the small θ

limit where Eq. (6.61) is valid, the detuning effects will predominate over the coherence loss mechanisms.

In this regime, the measurement interaction will not be giving information about the system dynamics

of interest, but only of the dynamics of the detuned system. This observation is consistent with the

behaviour demonstrated in Fig. 6.3 where we considered regular atom injections. There, we noted that

the small θ, high measurement rate (γ) limit generated a detuned slowly decaying oscillation to the

system of interest. We further note that, for θ small, the detuning is δ is consistent with Eqs. (6.56)

and (6.28), and that the measurement caused loss of decay Γ is consistent with Eq. (6.56).

The small θ behaviour is well illustrated in Fig. 6.5 where we vary the average probe atom arrival rates

indicated by β in units of the free evolution rate κ. The free evolution of the Rydberg atom-cavity mode

system is indicated for β = 0. As β is increased, we see a marked detuning of the system, with only very

weak decay of coherence as a result of the measurement interaction. This is not a good measurement

regime in which to obtain information about the evolution of the system.

A plot of the survival probability versus time for the case of Poisson injected probe atoms,

with different injection rates β for θ = 0.025 π.

Figure 6.5: Detuned small θ evolution for Poisson arrival times

We must turn to the full master equation (6.59) to determine whether there are regimes, for large θ

in which an effective measurement of the system of interest can occur.


Weak measurement limit - small β

The above derivations are valid in the large γ limit. Here, we consider the case where γ ¿ κ and so we

have β ¿ 1. In this case the rate of probe injection is much less than the time scale of coherent evolution

of the system away from the initial state. The eigenvalues of the dynamics are approximately

λ1 = −β sin2 θ

2

λ2,3 = −β

2sin2 θ

2± i. (6.63)

The survival probability exhibits a slowly damped oscillation as can be seen in the small β regime

of Fig. 6.5. (This same behaviour can also be seen in the same regime in the yet to be introduced

Fig. 6.6). In the case of this limit occurring with θ small, the eigenvalues, in dimensioned units, are

λ1 = −γθ2/4, λ2,3 = −γθ2/8 ± iκ which are consistent with the regular injection result in the same

limit. This behaviour was demonstrated in Fig. 6.3.

Strong measurement limit - large β, large θ

In this section we consider the case of a very large rate of injection of probe atoms, such that κ ¿ γ and

therefore β À 1. We consider this limit for strong probe atom monitoring with θ large. For β À 1 the

approximate eigenvalues of the dynamics are

λ1 ≈ 0

λ2,3 ≈ −β sin2

(θ

2

)± i

β

2sin(θ). (6.64)

These results give some indication of the relative strengths of the detuning and measurement terms

in the large β limit. Consider for example the case θ = π/2 where we have λ2,3 ≈ β/2 (1 ± i) and

we see that the rate of coherence loss and the measurement generated detuning magnitude are roughly

commensurate. The special case of θ = π is worthy of comment. In this case the approximate eigenvalues

of the evolution as given above are λ2,3 = −β with no detuning effects. This suggests that this regime

would be a good measurement regime, however, as we show below this matter is more complicated than

this.

We must now turn to the question of determining the “good measurement” regime. A considerable

part of the emphasis of Chap. (5) was that we must demonstrate the existence of a good measurement

regime, rather than simply establishing that a mechanism for coherence loss exists. We did this in the

previous chapter by asking whether the coherence loss mechanisms channelled information about the

atomic level populations from the microscopic realm to the macroscopic realm. We found some regimes

where this did occur effectively, and others where it did not. In this section we will find that we must

answer the same question.

We employ Eq. (6.42) to give guidance on the effective measurement regimes. Inspection of this

equation shows that the information obtained by a selective measurement interaction goes as 〈sin(θa†a)〉.


In turn, this demonstrates that with zero or one photon in the cavity, an experimenter obtains the best

information when θ = π/2 while no information is obtained for θ = π.

We must distinguish between the obtaining of information constituting a measurement and the action

of coherence loss mechanisms on the system. We established in the previous chapter, that there were

regimes where good photon statistics did not effect an atomic level measurement. This is demonstrated

again with the interaction being considered here. Here again we must consider the presence of a channel

for information flow from the system to the macroscopic realm in order to establish whether an effective

measurement interaction is occurring.

A more realistic model of the measurement interaction will also vary the interaction strength of the

injected atom and the cavity field. This will vary θ in an unpredicitable way. Once this is done, then

those system coherences that are not “killed off” for one value of θ will be “killed off” for the next.

This has been demonstrated by Brune et al. [82, 83]. Of course, as we have shown, significant detuning

can also be a result of the interaction. In this section we are modelling the injection of the atoms as

being Poisson distributed, but keeping the interaction strength as being fixed. However, in Eq. (6.43)

we showed that, the variation in the interaction strength does not modify the average ionization rate,

and thus, the variation in θ will not change our above conclusions.

In what follows we explore various values of θ to examine the dynamics of the system. In particular we

seek to examine the relation between the coherence loss mechanisms and the detuning mechanisms. The

first interesting case is when θ = π. We know that we have significant coherence loss, and no detuning of

the system, though we are not effecting a measurement. We can write the master equation [Eq. (6.59)]

exactly as

˙ρ = −iκ[σx, ρ] − γ [σz, [σz, ρ]] , (6.65)

which is identical to the Angular Momentum model introduced previously. However, in this system we

must not interpret this equation as being a measurement of the system populations. The eigenvalues of

the dynamics of the Bloch vector [Eq. (6.60)] are exactly given by

λ1 = −β

λ2,3 = −β

2± i

√1 − β2

4. (6.66)

The dynamics are characterised by a transition from underdamped to overdamped motion in the case

of β > 2. The effect of the probe atoms on the effective two level dynamics is equivalent to pure phase

decay. Indeed the sequence of probe atoms is equivalent to a random sequence of phase reversals in the

two level density matrix as discussed in the previous section.

In Fig. 6.6 we solve Eq. (6.60) exactly for θ = π and for θ = π/2. The free evolution of the Rydberg

atom-cavity mode system is seen for β = 0, while the small β behaviour described above is also apparent.

For large β we see that the system undergoes a steady decay of the initial state occupancy probability


while the similarity of the figures demonstrates the dominance of the coherence loss terms over the

measurement detuning terms.

This behaviour, for both θ = π and θ = π/2 is typical of measurement affected dynamical behaviour,

as is the decay of the two level occupancy probability to a long time value of 1/2, and is a result of

coherence loss. However, we can only interpret the interaction as a measurement in the case where

θ = π/2. In this case, the rate of decay of the initial state occupancy is inversely proportional to the rate

of probe atom injection, which is also typical of measurement induced dynamics and demonstrates the

quantum Zeno effect. Strictly, we are only able to claim that a quantum Zeno effect has been induced

in the case θ = π/2 as we have defined this effect as being measurement induced.

These results are readily demonstrated by considering the rate equation solution to the equations of

motion for the Bloch vector [Eq. (6.60)]. For large β we can set X = Y = 0 giving Z = −Z/β. These

equations are then solved to give

X(t) = − 1β

Z(t)

Y (t) = − 1β

Z(t)

Z(t) =12e−t/β . (6.67)

These equations demonstrate the above mentioned dependencies of the initial rate of decay of the occu-

pancy probability [related to Z(t)], and the expected long time behaviour. This rate equation approach

will be further discussed in Chap. (7) where we consider in more detail the rate equation solutions of

the full Angular Momentum model master equation.

6.5.3 Physical regimes

Is the strong measurement limit β À 1 achievable? The maximum size of γ is immediately given by the

requirement that we have only one atom in the cavity at a time. If the length of the cavity is Lc and

the velocity of the atoms is v then γ = v/Lc . We thus require an object atom with single atom Rabi

frequency κ such that κ ¿ v/Lc. However θ = χτ/2 is proportional to the time of flight of the atom

through the cavity, which cannot be greater than γ−1 without violating the assumption that there is

only one atom in the cavity at a time. Increasing γ will necessarily decrease θ which must be balanced

by large χ.

Typically single photon rabi frequencies for Rydberg atoms are in the range [84] 104 s−1 < κ, χ <

105 s−1. Thus we require γ > 105 s−1. For a 1cm length cavity this means a velocity of the order

of 1000m/s. The velocity of probe atoms is limited by the source. It could possibly be increased by

reversing a laser cooling scheme, i.e propagating a resonant laser field along the beam of probe atoms.

For example using a scheme of two slightly detuned counter propagating waves [86], velocities of the

order of 1000m/s could be achieved for sodium.


A plot of the survival probability versus time for the case of Poisson injected probe atoms,

with (a) θ = π (b) θ = π/2. The graphs are similar as the coherence loss terms dominate the

detuning. We note that the case of θ = π cannot strictly be interpreted as a measurement for

fixed θ.

Figure 6.6: The initial state survival probability for various θ.

6.6. CONCLUSION 113

With this value of γ we require θ ≈ π/2 and this implies that χ ≈ π105s−1 close to the regime quoted

above. This suggests that this proposed mechanism might be feasible.

6.6 Conclusion

In this chapter, we have again demonstrated the main themes of this thesis. Namely, that any analysis

of a measurement must employ an explicit model of the measurement interaction in order to address the

specific question of whether a measurement of the system dynamics of interest is actually occurring. This

analysis must treat the detuning caused to the system due to its being embedded in the larger combined

system-detector interaction, and then demonstrate that a measurement is being accomplished through

an examination of some nonunitary interaction mechanism that provides a channel for information flow

from the microscopic realm to the macroscopic realm.

In this chapter we treated photon number measurements employing a beam of Rydberg atoms. We

also treat photon number measurements in the next chapter, but effect this measurement interaction by

using a quadratic coupling of cavity modes based on four wave mixing.

rydberg-atom phase sensitive detection - uq espace157843/n07chapter6.pdfrydberg-atom phase sensitive...

Documents