1. Phase Selection in Interference of Non-Classical Sources
Advanced Methods in Plasma and Optics
In honor of Amnon Fisher’s 70th birthday
Phase Selection in Interference of Non-Classical Sources
Good evening, my name is Ofer Firstenberg. I’m working under the supervision of Amiram Ron… as part of Amnon’s group.
And I’m going to tell you a little about interference of two sources mainly in quantum optics.
Ofer Firstenberg, Yoav Sagi, Moshe Shuker,
Amit Ben-Kish, Amnon Fisher, Amiram Ron
Department of Physics, Technion - Israel Inst. of Tech.

2. Outline The chronicles of two-source interference
A generic two-source interference system and its oscillating “phase state”
A scheme for quantum-non-demolition (QND) measurement of interference.
Simulating the emergence of oscillating states.
Conclusions
The outline of the talk is as follows:
I’ll review past experimental and theoretical work on two-source interference, including recent work on Fock-state interference.
I will present a simple generic system of two-source interference for which the interesting quantum states can be easily investigated. A special state is what we call: “The oscillating phase state”, which is highly non-classical but exhibit many classical features, including 100% visibility oscillations.
In the main part of this talk I’ll introduce a new sophisticated experimental scheme, in which two-source interference can be measured using Quantum-Non-Demolition measurements. I’ll review some several important advantages of this scheme over conventional ones.
Finally, I’ll present some Monte-Carlo simulations of experiments with the proposed system, and I’ll show how the oscillating phase state emerges in all cases.

3. Observations of Two-Source Interference
“Each photon interferes only with itself. Interference between two independent photons never occurs” Dirac, 1930
1949: Independent microwave beams (Hull)
50’s: Incoherent light (Forrester; Brown & Twist)
60’s: Independent lasers
Temporal (Javan et. al.)
Spatial (Magyar & Mandel)
Attenuated lasers
(Paul et. al.; Radloff)
In 1930, in his famous textbook on quantum mechanics, Dirac made a basic statement on optical interference:…
We can nowadays safely say that this was not precise. Interference between two independent sources has been observed since the late 40’s:
In 1949 Hull has observed transient temporal interference between two independent microwave sources.
In the 50’s, Famous experiments of incoherent light mixing where conducted by Brown& Twist, and by Forrester.
In the 60’s, the interference between two independent optical lasers has been observed, both temporal and spatial.
Later on, experiments has been done with attenuated laser beams, in which it was shown that only a single photon quanta live inside the system in any given time.
However, as it was proven by Glauber for incoherent thermal light and by Paul for attenuated lasers, The classical description applies equally well to all these experiments. Quantum mechanically, that means that they could all be described using some simple mixture of Glauber’s Coherent states.

4. Non-Classical Sources Interference
Spontaneous emission from two atoms (Dicke, Richter) or more (Fano, Mandel)
Late 80’s: Observation of two photons interference using PDC (Mandel, Franson)
“…The two radiating atoms could be extremely far apart … and still exhibit this correlation effect. … It should be remembered, however, that both atoms are coupled to the same electromagnetic field. In the process of emitting the first photon, this common coupling results in the excitation of correlation states between the two atoms.” Dicke, 1964
Non-classical states in the form of Number states, in which we are interested, were not considered back than.
The only non-classical states that were investigated were atomic sources. The case of two excited atoms, that are fixed in certain position, which are together able to emit two photons was studied thoroughly, with the well known result that the intensity correlation from two detectors exhibit non-classical interference.
For example, two detectors can be positioned in front of two excited atoms, such that they will never click together. This means that after the first photon was emitted from one of the atoms and was detected, the other atom somehow knows that.
As Dicke commented,…
In the next 20 minutes I will show you that it is not required that there will be any emission at all. The common measurement is sufficient.
Nevertheless, back then, there where no experimental capabilities to test the two-atoms case. The validation of the theory of two-photon interference was done using Parametric Down Conversion sources by Mandel, Franson, and others, some twenty years ago.
//Late 70’s: Proven by Richer that two-photons interference has analogy between many-excited atoms to a Number states fields (Richter)

5. Fock State Interference |ψ0=|N a| Nb
Y-T. Chough, PRA 55, 3143 (1997).
K. Molmer, PRA. 55, 3195
Expectation values read no interference.
Trajectory formalism show interference:
Continuous damping subjects non-unitary evolution
Photon detections described by “jump” operators
Environment modes are ignored.
phase is chosen randomly.
In the past decade, several authors have addressed the question of Fock state interference:
For example, This system was presented by Chough. Here, two single-mode cavities are positioned in front of a screen, and photons are allowed to leak out and be detected. This is a sort of variation of Young’s two-slits experiment of spatial interference.
The initial state of each cavity is a Fock state – A state for which the number of photons is definite.
Another system was suggested by Klaus Molmer in the same volume of Phys. Rev. A. This is a variation of the Homodyne detector to measure temporal interference. Here, two slightly-detuned cavities shed there photons on a beam-splitter and two detectors read the intensity behind it.
If we follow our classical intuition, for example in this case [left]. In classical Young interference, the position of the fringes on the screen depends on the relative phase between the waves in the two slits.
However, an initial Fock state do not have any definite relative phase in it. In fact, since the photon number is fixed, it is argued that the uncertainty of phase has the maximum value of two pi.
For this reason, conventional expectation values for initial Fock state give zero interference… This argument implies equally well for the temporal interference.
However, using numerical simulations, employing what is called the Trajectory Formalism, both authors have shown that in each single trajectory, or single realization, of the simulation, interference fringes or beating do appear, in both cases with 100 percent visibility. The fringes location, and hence the relative phase, is randomly chosen.
This trajectory formalism is a very useful mathematical tool for solving master equations, but it was never proven that each of its trajectories actually describes reality, or a possible outcome of experimental realization.
It has several non-orthodox, sometimes disputed, aspects:
Since environmental modes are ignores in the analysis, the continuous damping requires the creation of an effective Hamiltonian, which is not Hermitian, and which subjected the wave function to non-unitary time-evolution.
Each photon detection cause the system to “Jump”, according to the operation of some “jump” operators.

6. BEC Interference - Initial state is disputed -
Y. Castin, J. Dalibard, Phys. Rev. A, 55, 4330 (1997).
M.R. Andrews, C.G. Townsend, H.-J. Miesner, D.S. Durfee, D.M. Kurn, W. Ketterle, SCIENCE 275, 637 (1997).
- Initial state is disputed -
J. Javanainen, S.M. Yoo, Phys. Rev. Lett. 76, 161 (1996).
To complete the picture, I would like to show some results from the BEC area. Theoretical analysis by Javanainen and Yoo, and by Castin and Dalibard, of interference between two Bose condensates had shown similar results to the two cavities case.
And indeed, in 1997, Ketterle’s group had photographed an interference pattern of two condensates. Whether or not the initial state in the experiment was “Two Independent Fock states” is still disputed. If it was indeed in that Fock state, it would have been the only experimental observation of Fock state interference, up until now.

7. A Generic Two-Source Interference System
Source A
Linear
Superposition
Intensity
detectors
Source B
I’d like to begin by analyzing a simple generic system of two sources, Here they are, A and B. They can be thought of as simply two quantum harmonic oscillators.
With mean frequency OMEGA and frequency difference, or detuning, DELTA. Their Hamiltonian is simply this, with A and A-DAGGER being the annihilation and creation operators of mode A and B and B-DAGGER for mode B.
Now, in any interference system, these modes are somehow superimposed, eventually to be detected by one, two or more intensity detectors. In order to measure interference, one always have to measure the intensity in one or more “composed” modes.
Y. Sagi, O. Firstenberg, A. Fisher, A. Ron, Phys. Rev. A. 67, (2003).

8. A Generic Two-Source Interference System
Source A
Canonical transformation
Source B
However, to keep is simple and general, I will now ignore any such couplings or measurements. What i am doing, is just defining two new operators, C and D, by some rotation of A and B. C and D are again annihilation operators of harmonic oscillators, and writing the Hamiltonian in the new basis
The Hamiltonian written in terms of C and D turns out to be of two linearly coupled harmonic oscillators. As it is well known classically, in a system of coupled harmonic oscillator the energy oscillates between the modes.
Using that Hamiltonian, it easy to calculate the time dependence of the expectation value of Nc, which is the mean photon number in the composes mode C, or - the intensity in mode C. It is clear that Nc oscillates, and depending on the initial quantum state, may exhibit oscillations with 100% visibility.
Actually, in a former work, we have shown that this exact term is the intensity expectation value in Momler’s Homodyne system, if we do take into account all environment’s modes. But for now, we keep it simple. .
Y. Sagi, O. Firstenberg, A. Fisher, A. Ron, Phys. Rev. A. 67, (2003).

9. States of the Composed Modes
Coherent State
Fock State
Now, lets examine several interesting quantum states of these composed modes.
First, the Glauber’s coherent states. We use this diagram representation of state, where this axis is the number of photons in mode C, this is the number of photons in mode D, and color represents the absolute value of the amplitude of the Nc-Nd Fock term. The coherent state looks like this…
A well known result of Glauber’s is that in a coupled harmonic oscillators system, coherent state stays coherent all the time. So starting from a coherent state with a mean value of N photons, the dynamics looks like this….
This state exhibit 100% visibility oscillations in the intensity expectation value, while the number uncertainty in (in the sense of the uncertainty principle) the composed modes, oscillates between zero and square-root of N.
We now turn to a different initial state: The two independent Fock state of the original modes, A and B. This state looks like this in the composed mode representation, and it exhibit no expectation value oscillation. This trivial result is due to the fact that this state is an eigenvalue of our Hamiltonian. The number uncertainty in the composed modes for this state is of the order of N, much larger than the previous, square-root of N.

10. States of the Composed Modes
Coherent State
Fock State
Fock state in the composed mode
Finally, let me examine a special initial state: A Fock state in the composed modes. I start with N photons in mode C and vacuum in mode D. So, this is what happens: surprisingly or not, the state oscillates between C and D, becoming Fock state in D and then returning to be Fock state in C etc. etc.
The intensity expectation value is identical to the coherent state case. Actually, the number uncertainty is again oscillating between zero and square-root of N.
We call this special state: The oscillating phase state.
Oscillating “phase state”

11. The oscillating “phase state”
Definite total photon number
100% visibility oscillation, with
Does the system evolve towards that kind of state in the Fock interference experiments?
//Further analysis enables us to explicitly write the oscillating phase state. This state depends upon the phase parameter phi, And it has a definite total number of photons.
As we have seen it exhibit the same expectation value and number uncertainty as a coherent state, so prima facia, we will not be able to distinguish him from a coherent state if we only measured the intensity oscillations.
This leads me to a Poem, by the novice poet Molmer, how wrote
And we ask, accordingly…
(Molmer, 1997)

12. A Scheme for QND Measurement of Interference using Cavity QED
Field w D 
|e
|g w0
Atom
Perfect mirrors (lossless)
Atoms as detectors
Let me introduce a new experimental scheme for two-source interference, which is quite different from what we have seen so far.
The system contains two cavity modes that have some spatial overlap. This can be easily imagined as two cavities with orthogonal main axes, as is pictured here, but is probability more conveniently implemented using a double-mode cavity. The mode are slightly detuned, by the amount DELTA.
Now, our detectors will actually be two-level atoms that will pass through the overlap region. The detuning between the atomic resonance frequency and the cavity frequency is THIS DELTA.
In our analysis we follow the extensive work of Haroche and Co-Workers, who are doing many beautiful things with such Cavity QED system, using a single mode.
Two cavities (or single cavity with two nearly-degenerate modes)
Spatial overlap
 S. Haroche, J.M. Raimond, Advances in Atom. Molec. & Opt. Phys. Supplement 2, p. 123 (1994).

13. Off-Resonance Coupling ( « )
Negligible absorption probability (QND).
Light shift and Lamb shift.
Spatial
overlap
We are working in the Off-Resonance regime. A simplified version of the coupling Hamiltonian is written like this. In writing this we have already assumed that the spatial configuration of the electromagnetic modes is identical, at least where the atom pass. That way, the atom is coupled to A and B in identical way.
In the off-resonance regime, there is neglected probability to photon absorption, and in that sense the interaction may be called Quantum-Non-Demolition… It means that in average, energy does not leave the cavity.
However, the interaction does induce a shift of the atomic resonance - light and lamb shifts. And we see that is it proportional to the intensity in a composed mode.
//If we didn’t had a spatial overlap between the modes, the phase shift was proportional to the sum of intensities in both cavities. But as the overlap increases, the “each way” information, the knowledge of the atom with which mode it was coupled, is loosed, and the phase shift become proportional to the intensity in a compose mode.

14. Ramsey Interferometery
Transforming phase difference to excitation probabilities… 1
0.5
“g” Probability
“e” Probability
How do now measure the information stored in the phase? Using standard Ramsey interferometery.
Each atom is prepared, say, in the ground state, and in the first Ramsey zone it is rotated to a superposition of G and E.
When it pass through the cavity, the different part of the wave-function acquire different phase shift.
After leaving the Cavity, the atom is again rotated in a second Ramsey zone, and the excitation probability are a function of the phase difference, and hence a function of the photon number.
The parameters of the system can be chosen such that the excitation probability look like this.
This procedure is repeated many times, with more and more atoms. With each detection we learn more about the actual excitation probability, and thus learn more about the intensity in the composed mode.

15. The Bernoulli Trial Process
Each atom improves the estimation of intensity. Uncertainty of the estimation after K atoms is
Same result was obtained for photo-detectors
Effective detection (maximum of ) decreases our uncertainty to after atoms.
“Amount of information” in a single atom, determined by the interaction strength and duration.
The whole process is similar to the Bernoulli trial process, in which we flip an unbalanced coin, K times, in order to estimate the unbalanced probability.
Each atom improves the estimation of intensity, and it is well known that the estimation uncertainty decreases as the square-root of K.
We are able to control the “Amount of information” stored in a single atom by increasing the interaction strength.
When using the maximum “Amount of information” the uncertainty decreases to the square-root of N after about K equals N atoms.
 B.C. Sanders et. al., Phys. Rev. A. 68 (4), (2003)

16. Simulation: Initial Coherent state
~80 Atoms per cycle
Let me show you some examples of simulation results.
This is simulation with coherent initial state, with mean photon number 16. Remember that here we throw about 80 atoms in each oscillation cycle.
You can see the affect of the atoms on the dynamics is not noticeable.
Here I draw versus time the real expectation value of Nc and Nd, and you can see that it has some noise, associated with the measurement procedure, but generally the oscillation is stable.
On the bottom is an histogram of the detection record. You can see that the value of Nc can easily be extracted from that record.
Dynamics is not affected by the measurement.
Detections follow the interference signal.

17. Simulation: Initial Fock state Fock Coherent
~80 Atoms per cycle
Fock
Coherent
The symmetric state evolves into an oscillating state.
Detections identical to the coherent case!
We are now getting to the important part.
The initial state is the independent Fock state, with exactly 8 photons in mode A and 8 photons in mode B.
What you will now observe, is how the first atom decrease the width of the state, gradually transforming it an oscillating state.
The trace of that state collapse is evident in the expectation value.
You see that the detection record exhibit same oscillations as for the coherent case, and naturally, the oscillation phase can be extracted from the record.

18. Emergence of the Oscillating Phase State
Atoms per cycle 3 16 80
405
1026
No Atoms
Atoms
Robust emergence of stable oscillations with 100% visibility.
State stabilized after atoms, when the uncertainty is
Oscillation phase is distributed uniformly.
I’m showing here a slightly more complex history. The red error-bars represent the uncertainty in modes C and D.
At beginning no atoms are sent, and it is obvious the state is constant, with large number uncertainty.
Than, the first atoms (about 16) start changing the state, and transform it to the oscillating phase state. The rest of the atoms just keep monitoring that signal.
Here we stop sending the atom, letting the system evolve with no perturbation, and you see that the state that had developed is stable and robust, oscillating with 100% visibility.
The bottom figure show the distribution of the selected phase from 500 runs of the simulations. This fits exactly to the uniform distribution between 0 and 2π.
Finally, we recall that the after K equals N atoms, the uncertainty should be reduced to square-root of N, that is the oscillating phase state should be constructed. And indeed, as we see here for a range of atom rate, from 3 per cycle to about 1000 per cycle, The oscillation stabilize after about 16 atoms.

19. Conclusions
Two Fock states will always show interference when the composed modes are measured:
Initial independent Fock state has large number uncertainty in the composed mode.
The decrease of the uncertainty induce an evolution to a stable oscillating phase state.
The measured intensity is random, and hence the relative phase.

20. Thank You.