Do we need quantum light to test quantum memory? M. Lobino, C. Kupchak, E. Figueroa, J. Appel, B. C. Sanders, Alex Lvovsky
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1 Do we need quantum light to test quantum memory? M. Lobino, C. Kupchak, E. Figueroa, J. Appel, B. C. Sanders, Alex Lvovsky
2 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
3 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
4 EIT for quantum memory
5 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 GHz absorption signal frequency scan
6 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, (2008).
7 EIT-based memory: Quantum case 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., ] NJP 11, (2009) B A C
8 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
9 EIT for quantum memory: state of the art Existing 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?
10 Why we need process tomography In classical electronics Constructing any complex circuit requires precise knowledge of each component s operation 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
11 Why we need process tomography 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
12 Quantum processes General properties Positive mapping Trace preserving or decreasing Linear in density matrix space E( $ ρ1+ $ ρ2) = E($ ρ1) + E($ ρ2) Not always linear in the quantum Hilbert space b g ( ) ( ) E ψ + ψ = E ψ + E ψ The superoperator Tensor such that for any input density matrix a f a f the output density tensor is ρout = E ρ lk lk nm in nm Characterizing the process means finding the superoperator E lk mn a f ρ in mn
13 Quantum process tomography. The approach Direct approach [Laflamme et al., 1998; Steinberg et al., 2005; etc.] Prepare a set of probe states {ρ 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(ρ i )} Any arbitrary state ρ can be decomposed ρ = λiρi Linearity E( ρ) = λie( ρi) 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
14 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
15 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) z 2 $ ρin = P$ ρ ( α) α α d α phase space Application to process tomography in 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 The good news E( $ ρin) P$ ρ ( α) E α α d α = z b g 2 phase space in Coherent states are readily available from a laser. No nonclassical light needed Complete tomography Science 322, 563 (2008)
16 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: P n Sounds like bad news W$ ρ( α) = P$ ρ( α) W 0 ( α) ( α) F HG α The solution [Klauder, 1966]: I KJ 2 n b g δ α Any state can be infinitely well approximated by a state with a nice P function by means of low pass filtering Artist s view of P-function
17 Example: squeezed vacuum Wigner function from experimental data Bounded Fourier transform of the P-function Regularized P-function Wigner function from approximated P-function
18 Practical issues The superoperator ρ out ρ in Finding for a given is complicated need the superoperator tensor nm such that Approximations Need to choose the cut-off point L in the Fourier domain E lk Can t test the process for infinitely strong coherent states must choose some α max There is a continuum of α s process cannot be tested for every coherent state must interpolate a f a f ρout = Elk nm ρin nm lk
19 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
20 Memory for light as a quantum process PRL 102, (2009)
21 Superoperator reconstruction The experiment Input: coherent states up to α 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, (2009)
22 Superoperator reconstruction: the result a f a f Superoperator in the Fock basis: ρout = E ρ lk lk nm in nm mm Shown: diagonal elements E kk of the process superoperator Each color: diagonal elements of the output density matrix for input m Zero 2-photon detuning 540 khz 2-photon detuning 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 PRL 102, (2009)
23 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
24 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
25 Squeezing in our experiment Pump laser 10W (560 nm) We need: PRA 75, (2007) A narrowband squeezed light source at the rubidium wavelength (795 nm) Ti:Sapphire laser 1.8 W (795 nm) Frequency doubler 700 mw (397.5 nm) Parametric amplifier (795 nm)
26 The parametric amplifier PRA 75, (2007) Uses a 20-mm long PPKTP crystal Resonant to 87 Rb 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 squeezed vacuum noise vacuum noise level
27 Chopping squeezed light into microsecond pulses Home-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 PRL 100, (2008)
28 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 reconstruction using the maximum-likelihood method density matrix J. Opt. B 6, S556 (2004)
29 Tomography of pulsed squeezed light Quadrature data Density matrix Wigner function 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, (2008)
30 Storage of squeezed vacuum PRL 100, (2008)
31 Storage of squeezed vacuum The setup Quadrature data Density matrix Wigner function Quadrature noise Maximum squeezing: 0.21±0.04 db Confirmed both by MaxLik and direct quadrature binning Squeezing observed in the retrieved state! PRL 100, (2008)
32 Test of process tomography Prediction with calculated superoperator Result of a direct experiment Fidelity = 0.996
33 Summary Network analyzer for quantum-optical processes By studying what a quantum black box does to laser light, we can figure what 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
34 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 (L min, α max, interpolation) Extend MaxLik methods to process tomography Extend to multimode case Investigate classic processes (a, a, beamsplitter, optical CNOT gate)
35 Thanks! The team (quantum memory + processes): Jürgen Appel ( Niels Bohr Institute) Eden Figueroa ( Max Planck Institute) Mirko Lobino ( Bristol) Dmitry Korystov ( University of Otago) Connor Kupchak Barry Sanders Ph.D. positions available
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