MEMORY FOR LIGHT as a quantum black box. M. Lobino, C. Kupchak, E. Figueroa, J. Appel, B. C. Sanders, Alex Lvovsky

Transcription

1 MEMORY FOR LIGHT as a quantum black box 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: 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, (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., ] B A C E. Figueroa, M. Lobino, D. Korystov, C. Kupchak and A. L., New J. Phys 11, (2009)

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 Not always linear in the quantum Hilbert space b g E ψ 1 + ψ 2 = E( ψ 1) + E( ψ 2) Example: decoherence but Always linear in density matrix space E( $ ρ + $ ρ ) = E($ ρ ) + E($ ρ )

13 Quantum process tomography. The approach Direct approach [Laflamme et al., 1998; Steinberg et al., 2005; etc.] Construct a set of probe states {ρ i } 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(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 M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders and A. L., Science 322, 563 (2008) 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 ρ = λiρi

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 I KJ 2 n b g δ α 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

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 M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders and A. L., Science 322, 563 (2008) The superoperator ρ out 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 ρ in 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 M. Lobino, C. Kupchak, E. Figueroa and A. L., PRL 102, (2009)

20 Memory for light as a quantum process M. Lobino, C. Kupchak, E. Figueroa and A. L., PRL 102, (2009)

21 Process 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?) M. Lobino, C. Kupchak, E. Figueroa and A. L., PRL 102, (2009)

22 Process 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

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: 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 J. Appel, D. Hoffman, E. Figueroa and A. L., 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

28 Data acquisition for homodyne tomography Quantum-state reconstruction using time-domain homodyne tomography density matrix Wigner function A. L., M. Raymer, Rev. Mod. Phys. 81, 299 (2009)

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

30 Storage of squeezed vacuum

31 Storage of squeezed vacuum The setup Quadrature data Density matrix Wigner function Quadrature noise Maximum squeezing: 0.21±0.04 db J. Appel, E. Figueroa, D. Korystov, M. Lobino, A. L. 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 INDTEAD OF EPILOGUE Quantum-state engineering at the two-photon level

36 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 a b1 1 + c2 2 of zero- one- and two-photon Fock states. a n n

37 Scheme parametric down-conversion (amplitude γ) α β weak coherent state inputs signal Suppose both detectors have fired simultaneously. What could this mean? Both photons come from down-conversion (amplitude γ 2 ) One comes from down-conversion, another from a coherent state (amplitude γα, γβ) Both photons come from coherent states (amplitude α 2,αβ, β 2 ) 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

38 Theory parametric down-conversion (amplitude γ) α β weak coherent state inputs According to calculations, the signal state is expected to be... ( 2 ) 2 α / 2 + αβ 0 + βγ 1 γ 2 ψ + signal 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)

39 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.

40 Results vacuum 0 one photon 1 superposition a0 0 + a1 1 two photons 2 superposition a1 1 + a2 2

41 Results vacuum 0 one photon 1 two photons 2 superposition a a 2 2

42 Results vacuum 0 one photon 1 two photons 2 superposition a a a 2 2

43 Thanks! Ph.D. positions available 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