Transcript Molmer Kalus

Quantum Information
with
Continuous Variables
Klaus Mølmer
University of Aarhus, Denmark
Supported by the European Union and
The US Office of Naval Reseach
Continuous variables:
• Collective variables for macroscopic atomic samples.
• Optical field variables - continuous wave fields !
Interaction:
• Dispersive: Phase shifts, Faraday polarization rotation
• Homodyne measurement on the fields
Goals:
• Precision probing, atomic clocks
• Squeezing, entanglement, cats and Fock states
• Teleportation, quantum memory, quantum computing
Outline
• Continuous variable physical systems
• Gaussian state formalism
• Three applications:
– Squeezing and entanglement
– Magnetometry
– Photons from fields
• Outlook
Continuous variables for two-level atoms
• Many atoms in (|↑>+|↓>)/√2  <Jx>=Nat/2,
<Jy>=<Jz>=0
Var(Jy)Var(Jz) = |<Jx>|2/4  binomial noise
(M.U.S.).
|↑>
|↓>
let pat = Jz/√<Jx>, xat = Jy/ /√<Jx>, [xat,pat]=i
harmonic oscillator degrees of freedom.
Normally,
ρ1 atom is quantum.
”Ground state” is Gaussian in xat,pa
Also without 100 % optical pumping!
Here, collective
pat, xat, are our
Quantum Variables !
Continuous variables for polarized light
x-polarized light has <Sx> = Nph/2, <Sy> = <Sz> = 0.
let pph = Sz/√<Sx>, xph = Sy/ /√<Sx>, [xph,pph]=i
harmonic oscillator ground state, Gaussian in xph,pph
Interaction
•
Dispersive atom-light interaction:
σ+ (σ-) light is phase shifted by |↑> (|↓>) atoms
 Faraday polarization rotation, proportional to <Jz>
Hint = g SzJz = к pat pph
|↑>
|↓>
Update of atomic state due to interaction with a
light pulse
Hint = g SzJz = к pph pat
pph unchanged
xph  xph + к t pat
pat unchanged
xat  xat + к t pph
xph is measured:
we learn about pat, we ”unlearn” about xat
Gaussian states
State characterized by
• vector of (x’s and p’s) with mean values m
• γ = matrix of covariances, γij= 2<(yi-mi)(yj-mj)>
P(y) = N exp(-(y-m)T γ-1 (y-m))
Gaussian states (m and γ) transform under xx, xp and pp
interactions (linear optics, squeezing), decay and losses.
Gaussian states (m and γ) transform under measurements
of x’s and p’s (Stern-Gerlach and homodyne detection).
Quantum case:
Heisenberg uncertainty limit on γ
Update of Gaussian atomic state due to interaction
with a light pulse
A 0
 (
0
I
)
 ´A  AF
(
)
 FA  F
Loss of light:
γph(1-ε)γph +ε I
Atomic decay:
γat(1-ηΔt)γat +2(ηΔt) I
 ´´A 0
(
0
I
)
Interaction with a continuous beam
Continuous  frequent probing
(weak pulses/short segments of cw beam):
Before interaction: optical state is trivial *
After interaction: state is probed or discarded
Differential equation for atomic covariance matrix
This is a non-linear matrix Ricatti equation.
*: not true for finite bandwidth sources
Application 1:
Squeezing and Entanglement
Atomic spin squeezing due to optical probing
L. B. Madsen and K.M., Phys. Rev. A 70, 052324 (2004)
For the simple atom-light example (binomial distribution):
d
Var ( pat )  2 2 Var ( pat ) 2  Var ( pat )  1 /( 2  2 2t )
dt
Entanglement of two gases by optical probing :
Duan, Cirac, Zoller, Polzik
Measure (x1+x2) and (p1-p2)
Entanglement and The Swineherd
(Hans Christian Andersen 1805-1875)
” … when one put a finger into
the steam rising from the
pot, one could at once smell
what meals they were
preparing on every fire in
the whole town”
Entanglement costs:
”Ten kisses from the princess”
“Ask him,” said the princess, “if
he will be satisfied with ten
kisses from one of my
ladies.”
“No, thank you,” said the
swineherd: “ten kisses from
the princess, or I keep my
pot.”
How much entanglement can be generated over a
lossy optical transmission lines ?
Ans.: ”Any amount, with use of repeaters and distillation.”
What is the best possible entanglement, obtained with
Gaussian operations ? (Distillation forbidden).
Does the no-distillation theorem put an ultimate limit to the
entanglement that we can squeeze into a single pair of
distant oscillator-like systems ?
Direct transmission
x1 p1
x2 p2
I = I0 exp(-L/L0) = I0 (1-ε) , loss ε
Two-mode entanged state:
(x1-x2), (p1+p2)
EPR uncertainty: ΔEPR = (Var(x1-x2)+Var(p1+p2))/2 < 1.
ΔEPR  (1-ε) ΔEPR + ε ( 1, for large loss)
Entanglement of two gases, GEoF
loss
Polarisation
rotation
Polarisation
rotation
Faraday rotation with x-polarized beam
H = κ1p p1 + κ2p’ p2
= κ√(1-ε) p’ (p1+p2) - κ√ε pvacp1
Does the read-out
teach us more
about gas 1 or 2 ?
Atomic entanglement by probing is a bad idea !
Finite optical
squeezing
Transmitted beams
Optically probed atoms
A most surprising result:
Lars B. Madsen and K.M., to be published
Probed atoms (symmetric)
Infinitely squeezed light
Probed atoms (one way)
with squeezed light
with anti-squeezed light
squeezing: 0.1, 10
Finite entanglement, N=1/3 for arbitrary loss:
Teleportation fidelity, F = 1/(1+N) = 75 %
for unknown coherent state.
Application 2:
Quantum Metrology
Metrology with a quantum probe
A side remark
For a quantum physicist, it is natural to think of the time evolution of the
state as dynamical evolution of the physical state of the system,
e.g., as the wave function is getting narrow, ”the particle localizes
physically”.
According to the Copenhagen interpretation, however, the wave
function/density matrix/Wigner function is not the state of the
system, but rather a representation of our knowledge about the
system. When we measure, we learn something.
Complete* relationship with classical theory for the estimation of a
gaussian variable under noisy measurements (Riccati).
(*: but recall commutators and Heisenberg’s uncertainty relation)
Probing of a classical magnetic field
(K.M & L. B. Madsen,Phys. Rev. A 70, 052102 (2004) )
P(random signal | B)  P(B | signal)
Our SIMPLE approach:
Treat atoms AND light AND B field by a joint Gaussian probability
Covariance matrix for (B,xat,pat,xph,pph).
Analytical solution (no noise):
Long times: ΔB~ 1/(Nat t3/2)
• Independent of ΔB0
• not as 1/√Nat ,1/√t
Estimate a time dependent (noisy) magnetic field
Vivi Petersen and K.M. to be published
Noisy field
Estimator (mean of Gaussian)
Correct estimator for the current value of B(t) !
(See also Mabuchi et al.)
”Gaussian hindsight”
Noisy field
Estimator (mean of Gaussian)
1. 0.02 ms delay, minimizes
2. Temporal convolution
3. Gaussian distribution for (B(t1),B(t2),.. B(tn)
measurements now, update the past.
”If I knew then, what I know now”
Noisy field
Estimator (mean of Gaussian)
”Get wiser in 0.02 msec”
Application 3:
Leaving the Gaussian states
Photons from fields.
Leaving the Gaussian states
Gaussian states do not encode qubits
(two coherent states may be almost orthogonal,
but their superposition is not a Gaussian)
Gaussian states cannot be distilled (purified)
Photons from fields
Field or photon description of light?
Single mode: state and operator pictures equivalent.
Continuous beams, multi-mode:
Expansion on number states practically impossible.
Field picture useful: mean values, correlation functions.
Conditioned dynamics, collapses in Heisenberg picture?
From many to two modes:
OPO output:
Chose trigger and output modes:
Wigner function of four real variables:
(Gaussian, prior to click)

Click event (photodetection theory):
ρ  a1ρa1+
and then trace over mode 1
Wigner function transformation:
Gaussian
 Gaussian times a poynomial
Results:
(K.M., quant-pt/0602202, today)
Weak OPO
n=1 Fock state
Strong OPO
Schrödinger kitten
75 % detection
Experiments (Grangier group)
J. Wenger et al, PRL 92, 153601 (2004):
Experiments (Polzik Group, 2006):
(J. S. Neergaard-Nielsen et al, quant-ph/0602198)
Experiments
Weak OPO
65 % efficiency
Experiments
Strong OPO
65 % efficiency
Theory
Strong OPO
65 % efficiency
100 % efficiency
Conclusion/outlook,
• Many atoms and many photons are ”easy experiments”
(classical fields, homodyne detection)
• Many atoms and many photons is ”easy theory” (readily
generalized at low cost to more samples/fields)
• Gaussian states: squeezing, entanglement, … and also:
finite bandwidth sources, finite bandwidth detection
• Gaussian states unify quantum and classical variables:
classical B-field + atoms + light probe
 other observables: interferometry, … .
• First step to non-Gaussian states, discrete qubits,
Schrödinger Cats, … .