Embed Size (px)
Citation preview
MEMORY FOR LIGHT as a quantum “black box”
M. Lobino, C. Kupchak, E. Figueroa, J. Appel, B. C. Sanders, Alex Lvovsky
Outline
•
EIT and quantum memory for light•
Quantum processes: an introduction
•
Process tomography via coherent states•
Process tomography of quantum memory
•
Test with the squeezed state
Outline
•
EIT and quantum memory for light•
Quantum processes: an introduction
•
Process tomography via coherent states•
Process tomography of quantum memory
•
Test with the squeezed state
EIT for quantum memory
EIT in the lab•
Implementation in atomic rubidium•
Ground level split into two hyperfine sublevels
→ a perfect Λ
system•
Control and signal lasers must be phase locked to each other at 6.834 GHz
signal frequency scan
abso
rptio
n
EIT-based memory: in the laboratory
•
Practical limitations•
The pulse may not fit geometrically inside the cell
•
EIT window not perfectly transparent → part of the pulse will be absorbed
•
Memory lifetime limited by atoms colliding, drifting in and out the interaction region
•
In the quantum case: extra noise and decoherence
issues
•
Classical case: •
investigated theoretically and experimentally
•
Quantum case: •
not yet well studied
From N. B. Phillips, A. V. Gorshkov, and I. Novikova, Phys. Rev. A 78, 023801 (2008).
•
The extra noise•
Without decoherence, all atoms are in |B⟩
•
No extra noise•
With population exchange between |B⟩
and |C⟩,
•
some atoms move to |C⟩.•
They get excited into |A⟩
•
And re-emit into |B⟩
•
Spontaneous emission → quadrature noise in signal•
Not yet well studied [P. K. Lam et al., 2006-2008]
A
B
C
E. Figueroa, M. Lobino, D. Korystov, C. Kupchak and A. L., New J. Phys 11, 013044 (2009)
EIT-based memory: Quantum case
Outline
•
EIT and quantum memory for light•
Quantum processes: an introduction
•
Process tomography via coherent states•
Process tomography of quantum memory
•
Test with the squeezed state
EIT for quantum memory: state of the artExisting work
•
L. Hau, 1999: slow light•
M. Fleischauer, M. Lukin, 2000: original theoretical idea for light storage
•
M. Lukin, D. Wadsworth et al., 2001: storage and retrieval of a classical state•
A. Kuzmich
et al., M. Lukin
et al., 2005: storage and retrieval of single photons
•
J. Kimble et al., 2007: storage and retrieval of entanglement•
M. Kozuma
et al., A. Lvovsky
et al., 2008: memory for squeezed vacuum
= Various states of light stored, retrieved, and measured
An outstanding question•
How will an arbitrary
state of light be preserved in a quantum storage
apparatus?
In classical electronicsConstructing any complex circuit requires precise knowledge of each component’s operation
Why we need process tomography
This knowledge is acquired by means of network analyzers•
Measure the component’s response to simple sinusoidal signals
•
Can calculate the component’s response to arbitrary signals
•
In quantum information processing•
If we want to construct a complex quantum circuit, we need the same capability
•
Quantum process tomography•
Send certain “probe” quantum states into the quantum “black box” and measure the output
•
Can calculate what the “black box” will do to any other quantum state
Why we need process tomography
Quantum processes
•
General properties•
Positive mapping
•
Trace preserving or decreasing•
Not always linear in the quantum Hilbert space
•
Example: decoherence|1⟩
→ |1⟩
|2⟩
→ |2⟩
but
|1⟩
+ |2⟩
→ |1⟩⟨1| + |2⟩⟨2|
•
Always linear in density matrix space
E E Eψ ψ ψ ψ1 2 1 2+ = +b g ( ) ( )
E E E( $ $ ) ( $ ) ( $ )ρ ρ ρ ρ1 2 1 2+ = +
Quantum process tomography. The approach
•
Direct approach [Laflamme
et al., 1998; Steinberg et al., 2005; etc.]
•
Construct a set of “probe” states {ρi } that form a basis in the space of input density matrices (basis of the Hilbert space is insufficient!)
•
Subject each of them to the process•
Characterize each output {E(ρi )}
•
Any arbitrary state ρ can be decomposed •
Linearity
→
→ Process output for an arbitrary state can be determined
•
Challenges•
Numbers to be determined = (Dimension of the Hilbert space)4
•
Process on a single qubit
→ 16 •
Process on two qubits
→ 256
•
Need to prepare multiple, complex quantum states of light→ All work so far restricted to discrete Hilbert spaces of very low dimension
ρ λ ρ= ∑ i iE E( ) ( )ρ λ ρ= ∑ i i
Outline
•
EIT and quantum memory for light•
Quantum processes: an introduction
•
Process tomography via coherent states•
Process tomography of quantum memory
•
Test with the squeezed state
The main idea•
Decomposition into coherent states•
Coherent states form a “basis” in the space of optical density matrices
•
Glauber-Sudarshan
P-representation (Nobel Physics Prize 2005)
$ ( )$ρ α α α αρin P din
= z 2
phasespace
•
Application to process tomography•
Suppose we know the effect of the process E(|α⟩⟨α|)
on each coherent state
•
Then we can predict the effect on any other state
E E( $ ) ( )$ρ α α α αρin P din
= z b g 2
phasespace
•
The good news•
Coherent states are readily available from a laser. No nonclassical
light needed
•
Complete tomography
M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders and A. L., Science 322, 563 (2008)
ρ λ ρ= ∑ i i
The P-function [Glauber,1963; Sudarshan, 1963]
•
The problem•
P-function is a deconvolution
of the state’s Wigner
function
with the Wigner
function of the vacuum state
•
For nonclassical
states (photon-number, squeezed, etc.): extremely ill-behaved
Example: Pn
n
( )αα
δ α∝ −∂
∂FHGIKJ
2
b g
•
Sounds like bad news
•
The solution [Klauder, 1966]:Any state can be infinitely well approximated by a state with a “nice” P function by means of low pass filtering
W P W$ $( ) ( ) ( )ρ ρα α α= ∗ 0
Example: squeezed vacuum
Bounded Fourier transform
of the P-function
Regularized P-function
Wigner
function from experimental data
Wigner
function from approximated P-function
Practical issues•
The superoperator•
Finding for a given is complicated
⇒ need the superoperator
tensor such that •
Approximations•
Need to choose the cut-off point L in the Fourier domain
•
Can’t test the process for infinitely strong coherent states ⇒ must choose some αmax
•
There is a continuum of α’s ⇒ process cannot be tested for every coherent state
⇒ must interpolate
ρ ρout ina f a flk lknm
nm= EElknm
ρout ρin
M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders and A. L., Science 322, 563 (2008)
Outline
•
EIT and quantum memory for light•
Quantum processes: an introduction
•
Process tomography via coherent states•
Process tomography of quantum memory
•
Test with the squeezed state
M. Lobino, C. Kupchak, E. Figueroa and A. L., PRL 102, 203601 (2009)
Memory for light as a quantum process
M. Lobino, C. Kupchak, E. Figueroa and A. L., PRL 102, 203601 (2009)
Process reconstruction
•
The experiment•
Input: coherent states up to αmax
=10; 8 different amplitudes•
Output quantum state reconstruction by maximum likelihood
•
Process assumed phase invariant•
Interpolation
•
How memory affects the state•
Absorption
•
Phase shift (because of two-photon detuning)•
Amplitude noise
•
Phase noise (laser phase lock?)M. Lobino, C. Kupchak, E. Figueroa and A. L., PRL 102, 203601 (2009)
Process reconstruction: the result
•
Superoperator
in the Fock
basis: •
Shown: diagonal elements of the process superoperator
•
Each color: diagonal elements of the output density matrix for input |m⟩Ekk
mm
•
How can we test if this is correct?•
Store, retrieve, and measure a nonclassical
state of light
•
Calculate the expected retrieved state from the superoperator•
Compare the two
ρ ρout ina f a flk lknm
nm= E
Zero 2-photon detuning 540 kHz 2-photon detuning
Outline
•
EIT and quantum memory for light•
Quantum processes: an introduction
•
Process tomography via coherent states•
Process tomography of quantum memory
•
Test with the squeezed state
How to produce squeezing?•
Non-degenerate parametric down-conversion•
Photons are different in direction, frequency, polarization
•
Used e.g. to create entanglement
•
Degenerate parametric down-conversion•
Photons are identical
•
If we can generate enough pairs, output will be squeezed•
Use optical cavity to enhance nonlinearity
Squeezing in our experiment
Pump laser 10W (560 nm)
Ti:Sapphire laser 1.8 W (795 nm)
Frequency doubler 700 mW (397.5 nm)
Parametric amplifier (795 nm)
We need:A narrowband squeezed light source at the rubidium wavelength (795 nm)
The parametric amplifier
•
Uses a 20-mm long PPKTP crystal•
Resonant to 87Rb absorption line
•
Oscillation threshold: 50 mW•
About 3 dB of squeezing
•
Squeezing bandwidth 6MHz•
Cavity length actively stabilized with an auxiliary phase locked laser
•
Squeezing limited by grey tracking
J. Appel, D. Hoffman, E. Figueroa and A. L.,
PRA 75, 035802 (2007)
vacuum noise level
squeezed vacuum noise
Chopping squeezed light into microsecond pulsesHome-made mechanical chopper
•
Use an old hard disk•
Accelerate to 200 Hz
•
Attach a slit to outer rim (50 μm = 1 μs)•
Shutter open most of the time → we
can determine the optical phase
Duty cycle
Data acquisition for homodyne tomography
Quantum-state reconstructionusing time-domain homodyne tomography
→ density matrix → Wigner
function
A. L., M. Raymer, Rev. Mod. Phys. 81, 299 (2009)
Tomography of pulsed squeezed light
Quadrature data Density matrix Wigner function
•
-1.86 dB of squeezing and 5.38 dB of antisqueezing•
Some squeezing lost due to time-domain tomography
•
This is the “initial state” we want to store
Storage of squeezed vacuum
Storage of squeezed vacuum
Quadrature noiseDensity matrix Wigner function
Quadrature data
•
Maximum squeezing: 0.21±0.04
dB J. Appel, E. Figueroa, D. Korystov, M. Lobino, A. L.
PRL 100, 093602 (2008)
The setup
Test of process tomography•
Prediction with calculated superoperator
•
Result of a direct experiment
•
Fidelity = 0.996
Summary
•
“Network analyzer” for quantum-optical processesBy studying what a quantum “black box” does to laser light, we can figurewhat it will do to any other state•
Complete characterization
•
Easy to implement
•
Application to quantum memory for light •
Full experimental characterization of quantum memory
•
Verified by storing squeezed vacuum
Outlook•
Quantum memory for light•
Develop full quantum theoretical understanding of EIT-based memory
•
Store quadrature entangled states•
Try different storage media and methods
•
Quantum process tomography•
Better understand the practical issues (Lmin
, αmax
, interpolation)•
Extend MaxLik
methods to process tomography
•
Extend to multimode case•
Investigate “classic” processes (a, a†, beamsplitter, optical CNOT gate)
INDTEAD OF EPILOGUE Quantum-state engineering at the two-photon level
Motivation
•
The ultimate vision •
Be able to produce and characterize an arbitrary quantum states of the light field
•
Existing achievements•
Squeezed [Konstanz] and quadrature entangled [Caltech,…] states
•
One-
[Konstanz] and two-
[Paris] photon Fock
states •
Single-
and dual-rail qubits
[Konstanz]
•
Photon-added states [Florence]•
“Schrödinger kittens” [Paris, Copenhagen, Tokyo]
•
What we report•
Arbitrary superpositions
of zero-
one-
and two-photon Fock
states.
∑ nan
210 210 cba ++
Scheme
Suppose both detectors have fired simultaneously. What could this mean?•
Both photons come from down-conversion (amplitude ∝ γ2)
•
One comes from down-conversion, another from a coherent state (amplitude ∝ γα, γβ)•
Both photons come from coherent states
(amplitude ∝ α2 , αβ, β2)
These possibilities are indistinguishable!⇒ By choosing coherent state amplitudes and phases, one can generate any linear combination of zero-, one-
and two-photon Fock
states
weak coherent state inputsα β
parametric down-conversion (amplitude γ)
signal
Theory
•
According to calculations, the signal state is expected to be...
•
If β
= 0: no 1-photon component (Hong-Ou-Mandel effect on the first beam splitter)
•
If α
= 0: no 0-photon component (the photon on the first detector must come from down-conversion)
( ) 2102/ 22 γβγαβαψ +++−∝
weak coherent state inputsα β
parametric down-conversion (amplitude γ)
signal
Experimental issues•
Down-conversion amplitude γ•
Must be high enough so 2-photon events are reasonably frequent
•
Must not be too high so higher photon number contribution is insignificant→ In our experiment:
laser repetition rate 76 MHz, down-conversion in PPKTP, γ
~ 0.1.•
Coincidence count events: 20 s−1
or higher•
Fraction of 3-photon events: ~ 1%, i.e. negligible
•
Phase stabilization•
Local oscillator is the phase reference
•
Relative phase stability of the 2 coherent states is crucial→ Use calcite beam displacers to make the interferometer
•
Inefficient detection•
Mode mismatch between the signal and the local oscillator
•
Linear losses•
Electronic noise
→ Detection efficiency is 55%. We correct for it in the state reconstruction.
Results
vacuum |0⟩ one photon |1⟩ two photons |2⟩
superposition a0
|0⟩
+ a1
|1⟩ superposition a1
|1⟩
+ a2
|2⟩
Results
vacuum |0⟩ one photon |1⟩ two photons |2⟩
superposition a0
|0⟩
+ a2
|2⟩
Results
vacuum |0⟩ one photon |1⟩ two photons |2⟩
superposition a0
|0⟩
+ a1
|1⟩
+ a2
|2⟩
Thanks!
•
The team (quantum memory + processes):•
Jürgen
Appel
(→ Niels
Bohr Institute)
•
Eden Figueroa (→ Max Planck Institute)
•
Mirko
Lobino•
Dmitry
Korystov
(→ University of Otago)
•
Connor Kupchak•
Barry Sanders
•
The team (quantum state engineering):•
Nitin
Jain
•
Simon Huisman•
Erwan
Bimbard
Ph.D. positions availablehttp://qis.ucalgary.ca/quantech/
![QUANTUM MEMORY: DESIGN AND OPTIMIZATION Xiaotong Ni · quantum key distribution [BB84] in the early phase, as well as quantum metrology [GLM06] and quantum machine learning [LMR13]](https://img.pdfslide.tips/doc/110x75/5e1f5cc589c7e33bda676412/quantum-memory-design-and-optimization-xiaotong-ni-quantum-key-distribution-bb84.jpg)
