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Phys 531 Lecture 27 1 December 2005
Quantum Optics
Last time, discussed photon optics
Light energy comes in units of ℏω particles called photons observable with good detector
One consequence: light is intrinsically noisy
Today: introduce proper quantum theory Quantum field theory of light
Note, this work recognized with 2005 Nobel prize 1
Outline:
- Quantum fields
- Mode expansion
- Quantum states of light
- Photon tricks
Do need some QM for this Material won’t be on final
Warning: derivations today not rigorous, some definitions simplified Consult Scully and Zubairy for real deal
Next time: review... bring questions!
Quantum Field Theory
Want proper quantum theory for photon
From last time, expect E(r) ∼ wave function ψ(r)
Since:
- Probability ∝ |E|^2
- Polarization analogous to spin
- E exhibits interference like ψ
- Wave equation ∼ Schrodinger equation
3
Intuition is right, but one problem
Number of photons is indefinite: photons easy to create and destroy
“Normal” QM assumes fixed N N = number of particles
Problem not just that N is unknown:
Can have quantum system in superposition
|ψ〉 =
√^1
( |atom + photon〉 + |excited atom〉
)
Photon in superposition of existing or not!
Could measure this state with polarizer along x
Imagine sending repeated pulses: each pulse in state Ψ
Each time get either 100% or 0% transmission unpredictable per pulse
Works even if EX and EY are classical states with many photons Simple version of Schrodinger’s cat
Could also have superpositions of beam position, frequency, etc.
7
Mode Expansion
Still have problem that E = E(r)
How to handle Ψ[E(r))]?
Solve by decomposing field into modes Usually, mode = plane wave
Easier to work inside a box, volume V
Then only discrete values of k allowed Get sums instead of integrals
Main advantage: makes normalization easier
Can write any E(r, t) as
E(r, t) =
∑ k
Akei(k·r−ωkt)
so state of field uniquely specified by {Ak}’s
Each mode k = independent degree of freedom Do QM on each independently
Treat each Ak as independent quantum variable Now a simple variable like X for a particle
9
But still one issue: Ak is complex
Really
E(r, t) =
∑ k
Re [Ak] cos (k · r − ωkt)
− Im[Ak] sin (k · r − ωkt)
Real and imaginary part of Ak both needed really two quantum variables per mode
Define
qk =
√ 0 V 2 ℏωk
Re Ak pk =
√ 0 V 2 ℏωk
ImAk
Two main results:
(1) Energy eigenstates are En = ℏω(n + 12 ) for integer n
⇒ photons!
(2) q and p don’t commute: [q, p] =
i 2 for our scale factors
Then can’t know q and p simultaneously:
∆q∆p ≥ 1 4 13
What do q and p correspond to physically?
Have A ∝ q + ip = amplitude of mode
Field ∝ q cos(k · r − ωt) − p sin(k · r − ωt)
So q is amplitude of wave ∼ cos() p is amplitude of wave ∼ sin()
Call components “quadratures” of wave
Of course, arbitrary which is which usually imagine a “reference oscillator” ∼ cos Define modes relative to it
Quantum States of Light
Since ∆q∆p ≥ 1 /4, amplitudes can’t have definite values
Draw picture: Say 〈q〉 large, 〈p〉 = 0 and ∆q ∆p
E
t Amplitude of wave is uncertain
15
Or, if 〈q〉 large but ∆q ∆p:
E
t
Amplitude well-defined, but phase uncertain
Question: What if 〈p〉 and ∆q were large, and 〈q〉 and ∆p were small? Would the phase or amplitude be more certain?
What is uncertainty in N?
Have N ' q^2 + p^2
So ∆N '
( ∂N ∂q
) 2 ∆q^2 +
( ∂N ∂p
) 2 ∆p^2
1 / 2
[ (2q)^2 ∆q^2 + (2p)^2 ∆p^2
] 1 / 2
( q^2 + p^2
) 1 / 2
' N 1 /^2
since ∆q = ∆p = 1/ 2
Same quantum noise from last time!
19
If ∆q 6 = ∆p, called “squeezed state” Generally don’t have ∆N = N 1 /^2
Useful for precision interferometry:
- reduce noise in component measured
- increase noise in component not measured
Generate squeezed states in nonlinear crystals Rather difficult to achieve
Local expert: Olivier Pfister
Can have coherent state with α = 0
Then 〈N 〉 = 0: no photons present Called vacuum state
Electric field also zero on average But still have fluctuations ∆q = ∆p = 1/2:
E
t
Fluctuations called vacuum noise Help explain spontaneous emission 21
Another possible state: Fock state
Has definite energy (N + 1/2)ℏω → mode contains N photons
Both ∆q and ∆p large, = (^12)
2 N + 1
E
t
Note vacuum state is also a Fock state
Most common nonclassical source: “Parametric down conversion”
Use special crystal with nonlinear interactions
Can “split” one photon into two: ω 0 → ω 1 + ω 2 with ω 0 = ω 1 + ω 2 (conserve energy) k 0 = k 1 + k 2 (conserve momentum) k 1
k 2
k 0
25
Input pulse (ω 0 ) in coherent state
Produces output pulse of form
Ψout =
∑ n
cnΨn(k 1 )Ψn(k 2 )
where Ψn(k) is Fock state for mode k
Typically operate with c 0 ≈ 1, c 1 small, higher c’s negligible
Then usually get nothing, but sometimes get pair of single photons
Polarization depends on crystal setup
possible to produce state √^1 2
(X 1 Y 2 + Y 1 X 2 )
One way:
k 1 k 0
k 2
X Y
Y X
Crystals thin, can’t tell where photons emitted
27
Then say modes are entangled quantum states not separable
Gives interesting effects:
- Detect one photon, know that other is present Acts ≈ like perfect 1-photon source (don’t need entanglement)
- Violate Bell’s inequality
- Send messages immune to eavesdropping (quantum cryptography)
- Transfer arbitrary quantum state (quantum teleportation)
Before measurement, total field state is
√^1 2
(aX 1 Y 2 X 3 + aY 1 X 2 X 3 + bX 1 Y 2 X 3 + bY 1 X 2 Y 3 )
Rewrite by adding and subtracting many terms:
1 23 /^2
( aY 1 X 2 X 3 + aY 1 Y 2 Y 3 + bX 1 X 2 X 3 + bX 1 Y 2 Y 3 +aY 1 X 2 X 3 − aY 1 Y 2 Y 3 − bX 1 X 2 X 3 + bX 1 Y 2 Y 3
+aX 1 X 2 Y 3 + aX 1 Y 2 X 3 + bX 1 X 2 Y 3 + bY 1 Y 2 X 3
−aX 1 X 2 Y 3 + aX 1 Y 2 X 3 + bX 1 X 2 Y 3 − bY 1 Y 2 X 3
)
31
This factors into
1 23 /^2
[ (aY 1 + bX 1 )(X 2 X 3 + Y 2 Y 3 ) +(aY 1 − bX 1 )(X 2 X 3 − Y 2 Y 3 )
+(aX 1 + bY 1 )(X 2 Y 3 + Y 2 X 3 )
−(aX 1 − bY 1 )(X 2 Y 3 − Y 2 X 3 )
]
or 1 2
[ (aY 1 + bX 1 )ΨA + (aY 1 − bX 1 )ΨB
- (aX 1 + bY 1 )ΨC − (aX 1 − bY 1 )ΨD
]
When Bob makes measurement, wavefunction collapses
Then Alice’s photon in state related to Υ
Bob knows result of his measurement
Calls Alice on phone and tells her how to change her state to Υ (easy using λ/2 and λ/4 plates)
Now Alice has state Υ indistinguishable from original photon 3 33
If we could do this with 10^23 atoms instead of one photon, could teleport a mouse
Teleportation theory: C. H. Bennett et al. Phys. Rev. Lett. 70 1895 (1993)
Some experiments: D. Bouwmeester et al. Nature 390 575 (1997) T. C. Zhang et al. Phys. Rev. A 67 033802 (2003) M. D. Barrett et al. Nature 429 737 (2004)
