do we need quantum light to test quantum memory? · outline • eit and quantum memory for light...

Do we need quantum light to test quantum memory?

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

EIT for quantum memory

EIT in the lab

Implementation in atomic rubidium

Ground level split into two hyperfine sublevels

a perfect

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: Classical case

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

NJP 11, 013044 (2009)

EIT-based memory: Quantum case

Outline






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 components operation

Why we need process tomography

This knowledge is acquired by means of network analyzers

Measure the components response to simple sinusoidal signals

Can calculate the components 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

Linear in density matrix space

Not always linear in the quantum Hilbert space

The superoperator

Tensor such that for any input density matrix

the output density tensor is

Characterizing the process means finding the superoperator

Elkmn ina fmn

out ina f a flk lknm nm= E

E E E 1 2 1 2+ = +b g ( ) ( )

E E E( $ $ ) ( $ ) ( $ ) 1 2 1 2+ = +

Quantum process tomography. The approach

Direct approach [Laflamme

et al., 1998; Steinberg et al., 2005; etc.]

Prepare a set of probe states {i } that form a full 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(i )}

Any arbitrary state 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

= i iE E( ) ( ) = i i

Outline






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)

$ ( )$ in P din= z 2phasespace

Application to process tomography

Suppose we know the effect of the process E(||)

on each coherent state

Then we can predict the effect on any other state

E E( $ ) ( )$ in P din= z b g 2phasespace

The good news

Coherent states are readily available from a laser. No nonclassical

light needed

Complete tomography Science 322, 563 (2008)

The P-function [Glauber,1963; Sudarshan, 1963]

The problem

P-function is a deconvolution

of the states Wigner

function with the Wigner

function of the vacuum state

For nonclassical

states (photon-number, squeezed, etc.): extremely ill-behaved

Artists view of P-function

Example: Pnn

( )

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$ $( ) ( ) ( ) = 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

Cant test the process for infinitely strong coherent states must choose some max

There is a continuum of s

process cannot be tested for every coherent state

must interpolate

out ina f a flk lknm nm= EElknmout in

Outline






Memory for light as a quantum process

PRL 102, 203601 (2009)

Superoperator reconstruction

The experiment

Input: coherent states up to max

=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?)PRL 102, 203601 (2009)

Superoperator 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 |mEkk

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

out ina f a flk lknm nm= E

PRL 102, 203601 (2009)

Zero 2-photon detuning 540 kHz 2-photon detuning

Outline






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)

PRA 75, 035802 (2007)

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

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 m = 1 s)

Shutter open most of the time we

can determine the optical phase

Duty cycle

PRL 100, 093602 (2008)

Data acquisition for homodyne tomography

Oscilloscope

Segmented, time-domain

acquisition during the pulse

Integrate with the temporal

profile of the pulse

Normalize data using vacuum

state photocurrent

quadrature

values

Spectrum analyzer

Continuous, frequency-domain

acquisition

Spectrum analyzer is slow and cannot see the chopper

Amount of observed noise

depends on the optical phase phase values

Quantum-state reconstructionusing the maximum-likelihood method

density matrix J. Opt. B 6, S556 (2004)

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

PRL 100, 093602 (2008)

Storage of squeezed vacuum

PRL 100, 093602 (2008)

Storage of squeezed vacuum

Quadrature noiseDensity matrix Wigner function

Quadrature data

Maximum squeezing: 0.210.04

dB

Confirmed both by MaxLik

and direct quadrature

binning

Squeezing observed in the retrieved state! 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

, max

, interpolation)

Extend MaxLik

methods to process tomography

Extend to multimode case

Investigate classic processes (a, a, beamsplitter, optical CNOT gate)

Thanks!

The team (quantum memory + processes):

Jrgen

Appel

( Niels

Bohr Institute)

Eden Figueroa

( Max Planck Institute)

Mirko

Lobino

( Bristol)

Dmitry

Korystov

( University of Otago)

Connor Kupchak

Barry Sanders

Ph.D. positions availablehttp://qis.ucalgary.ca/quantech/

M. Lobino, C. Kupchak, E. Figueroa, J. Appel, B. C. Sanders, Alex LvovskyOutlineOutlineEIT for quantum memoryEIT in the labEIT-based memory:Classical caseSlide Number 7OutlineEIT for quantum memory:state of the artSlide Number 10Slide Number 11Quantum processesQuantum process tomography.The approachOutlineThe main ideaThe P-function [Glauber,1963; Sudarshan, 1963]Example: squeezed vacuumPractical issuesOutlineMemory for light as a quantum processSuperoperator reconstructionSuperoperator reconstruction:the resultOutlineHow to produce squeezing?Squeezing in our experimentThe parametric amplifierChopping squeezed lightinto microsecond pulsesData acquisition for homodyne tomographyTomographyof pulsed squeezed lightStorage of squeezed vacuumStorage of squeezed vacuumTest of process tomographySummaryOutlookThanks!

do we need quantum light to test quantum memory? · outline • eit and quantum memory for light...

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