Theoretical Quantum Optics - In Two Lectures

1. Theoretical Quantum Optics - In Two Lectures Barry C. Sanders iCORE Professor of QIS IQIS, University of Calgary http://www.iqis.org/ Quantum Information Summer School Waterloo, Ontario — 23-24 June 2004

2. Faculty Richard Cleve* (Comp Sci) D. Feder (Th. Physics) Peter Høyer* (Comp Sci) K.-P. Marzlin (Th. Physics) A. Lvovsky (Exp. Physics) Barry Sanders* (Th. Physics) J. Watrous (Comp Sci) Affiliates: D. Hobill (Gen. Rel.), R. Thompson (ion trap), R. Scheidler & H. Williams (crypto) Postdocs S. Ghose, H. Klauck, H. Roehrig, A. Scott, J. Walgate Students Iyad Abu-Ajamieh*, M. Adcock, S. Fast, D. Gavinsky,Gus Gutoski*, T. Harmon,Y. Kim, S. van der Lee,K. Luttmer, A. Morris, Xue-Song Qi*, Zeng-Bin Wang* Research Assistants L. Hanlen, Rolf Horn*, G. Howard Members of Calgary’s Institute for QIS * Attending this summer school

3. Electromagnetic Field Field Quantization Quantum States Transformations Atoms & Fields Sources & Detectors Successes of Q. Optics The Atomic Qubit The Photonic Qubit Entanglement & Correlation Squeezed Light & Continuous Variable Quantum Information Examples of Q. Optics in Q. Information Conclusions Outline I. Fundamentals & Methods II. Applications to Q. Information

4. I. Fundamentals & Methods

5. Introduction • Quantum Optics is the study of the electromagnetic field when quantum phenomena are manifest. • Some aspects of Quantum Optics concern the preparation of nonclassical states of light, manipulation and control of the quantum field, detection theory, protection against decoherence, and applications. • Many QI experiments have been realized via Quantum Optics experiments.

6. I.A Electromagnetic Field

7. Maxwell’s Equations • Electromagnetism is one of the four fundamental forces (others: gravity and strong & weak nuclear forces) • Dynamics governed by Maxwell’s equations. • Fundamental quantities: Electric field E, magnetic field H, and D(E), B(H). • In free space, D=0E, B=0H.

8. Vector Potential • Electric and magnetic field can be obtained from one vector potential A(r,t). • Restrict freedom of choice for vector potential by choosing a gauge; Coulomb gauge is common in quantum optics: • Use A(r,t) rather than both E(r,t) and B(r,t)

9. Wave Equation and Modes • The vector potential satisfies a (linear) wave equation: • Split vector potential into positive and negative frequency components and perform a mode decomposition. • Modes are orthonormal with respect to boundary conditions, and polarization k is a two-component vector.

10. Energy of the Free Field • Free field energy is given by a Hamiltonian corresponding to a sum of energies for modes, with each mode a harmonic oscillator. • The a’s are (dimensionless) Fourier coefficients, and the field quadratures are canonical coordinate/position for the mode.

11. I.B Field Quantization

12. Creation and Annihilation Operators • Field mode energy changes via frequency-dependent steps (photons) of • Achieved by replacing Fourier coefficients by matrices that satisfy commutator • Vector potential is now an operator: • Note: energy is not bounded, hence matrices are infinite-dimensional.

13. I.C Quantum States

14. Convenient Representations • Choose a representation based on ease of calculation and relationship to intuition (e.g. number of particles or coherence).

15. Can we create such states? • Number states and coherent state are ideals. • Typically we start with the vacuum state and amplify it or “squeeze” the vacuum fluctuations, or create quantum-correlated modes and prepare another by measurement and “post-selection”. • Strictly speaking, start with classical fields, fully describe processing, and obtain a density matrix  that best describes the field.

16. Other Useful States • Coordinate representation |q, which is an eigenstate of the coordinate operator • The squeezed vacuum state is • In the coordinate representation, the squeezed vacuum q| is a (complex) Gaussian function, with equal variances in both dimensions for nil squeezing (vacuum). • Number states in q-representation are the Hermite Gaussians.

17. Amplitude squeezed SQL Measured state Reality Check: Squeezed light Thanks to P. K. Lam’s group at ANU for these figures

18. Polarization States • The transversality condition arising from the gauge restriction ensures only two polarization states. • We can think about the single photon state |n=1 as a two-component vector • There is unfortunately confusion caused by this notation; e.g. |0 can represent a vacuum state or a single photon in the “0” polarization - generally discern from context.

19. Thanks to Rolf Horn (Calgary) for preparing this slide.

20. I.D Transformations

21. Types of Transformations • Practical (and interesting) Quantum Optics transformations are so far restricted to beam splitters, polarization rotation, and squeezing. • It is convenient to work in the coordinate (cosine quadrature) representation. • Such transformations preserve Gaussians. • Need non-Gaussian transformations to achieve universality - e.g. via nonlinear optics or measurement-controlled feed-forward.

22. Transformations • Squeezing: • BS: • Two-mode squeezing • Q. nondemolition interaction: • These transformations over k modes yield the Lie group Sp(2k,R); add displacement (mix with classical field) to obtain (Gaussian-preserving) [HW(k)]Sp(2k,R).

23. Resource: two-mode entangled state • Mixing a squeezed state and its antisqueezed counterpart at a beam splitter yields a two-mode squeezed state • In the limit 1, the two modes are perfectly correlated in x and x’. • This state is the entanglement resource in continuous variable quantum information.

24. Thanks to P. K. Lam’s group at ANU for these figures

25. The Kerr Nonlinearity • Remember that D=D(E). If D(E) is a cubic, then is attainable, and this is enough for universal transformations. • The problem is that D(E) is a power series with very low coefficients for higher-order terms. • Possible to use single-photon inputs, interferometry - SU(N) transformations - and post-selection via photodetection to obtain a probabilistic implementation of Kerr nonlinearity.

26. I.E Atoms & Fields

27. Two-Level Electronic System (Atom) • So far we have considered the field in isolation • The Jaynes-Cummings model describes coupling between a single (quantum) field mode and the simplest molecule/atom/dot, which has an electric dipole that is effectively a two-level quantum system. • Refer to “two-level atom” (2LA) for convenience. • Label levels as |0 and |1 (context distinguishes between number states, polarization states, etc). • Pauli operators:

28. Electric Dipole Coupling • In Quantum Optics, we treat the atom as an electric dipole (so the atom is small compared to the wavelength of the mode) and ignore rapidly oscillating terms in the Hamiltonian. • The classical field description emerges for strong fields and ; for an appropriate time of interaction, this Hamiltonian can create superpositions of |0 and |1.

29. electron nucleus orbital

30. I.F Sources & Detectors

31. Basics of Sources • The essentials of sources (and detectors) can be understood from studying electric dipole coupling between field and 2LAs. • Laser output can be modeled as sequence of excited 2LAs passing through optical cavity (region bounded by a mirror on each side). • Antibunched light from single atom - photon from de-excitation, then delay to re-excite.

32. Photodetection and Homodyne Detection • Fundamentally detection is excitation of 2LA. • Ideal photodetection would be a projective-valued measure corresponding to |nn|, but for single-photon counting module (SPCM) detect vacuum (no click) vs not-vacuum (click). • Array of efficient SPCMs converge to |nn|. • Ideal homodyne detection would be |xx|, but generally regard measurement as Gaussian distribution of |xx|.

33. Caveats • Ideal single-photon source works with efficiency p so, if we can ignore spread over other modes and higher-order photon numbers, the state is • An ideal laser produces a Poisson mixture of photon number, not a coherent state, but coherent state is a convenient descriptor of the output. • Squeezed light is never in a minimum-uncertainty state: the Gaussian wavefunction is convolved with a Gaussian noise function (not to mention that squeezed state is also a mixture of number states ).

34. I.G Successes of Q. Optics

35. Examples • Created nonclassical states of light including antibunched photons and squeezed light. • Wave-particle duality and complementarity. • Generated entangled states - as polarization entangled states or as two-mode squeezed state - and Bell’s inequality has convincingly been violated hence negating “local realism”. • Quantum information tasks such as quantum teleportation and sharing quantum secrets.

36. Theoretical Quantum Optics - In Two Lectures Barry C. Sanders Institute for Quantum Information Science University of Calgary http://www.iqis.org/ Quantum Information Summer School Waterloo, Ontario — 23-24 June 2004

37. II. Applications to Q. Information

38. Outline • The Atomic Qubit • The Photonic Qubit • Entanglement & Correlation • Squeezed Light & Continuous Variable Quantum Information • Examples of Quantum Optics in QI • Conclusions

39. Strengths of Quantum Optics for QI • Decades of technological expertise with generating, transmitting, and detecting qubits, ebits, and squeezed states. • Low decoherence during transmission. • Experiments are well understood: excellent agreement between theory and experiments. • New methods for enhancing Kerr-type nonlinearity via electromagnetic-induced transparency (EIT) and probabilistically via linear optics and detection+feed-forward.

40. II.A The Atomic Qubit

41. Realizing Quantum Gates Obtain NOT and Hadamard and other gates for appropriate t. Allows gate operations on two atoms. Direct dipole-dipole interaction allows two-qubit gates via coupling

42. II.B The Photonic Qubit

43. Two Types of Photonic Qubits • Dual Rail Qubit: A single photon can be split by a beam splitter to be in a superposition of going down either of two channels: |e corresponds to photon in channel • Polarization Qubit: A single photon is in a superposition of two polarizations: |0 may correspond to horizontal or left-circular or another polarization with |1 the opposite.

44. II.C Entanglement & Correlation

45. Creating Entanglement • Polarization-entangled states: parametric down conversion (PDC) converts a pump photon into a photon pair (in separate modes) of type |0|0 (Type I) or |0|1 (Type II). • Back-to-back Type I PDCs: |0|0 +|1|1. • Squeezed light: The two modes are entangled by quadratures and in number of photons. • [Some claim that the dual rail qubit is entanglement between photon in one channel and nothing (vacuum) in other channel.]

46. Correlations • Empirically observe entanglement via correlation measurements and checking that the results cannot be explained in terms of separable density matrix. • Polarization entanglement observed by detecting photon coincidences by local polarization rotations and obtaining correlations that violate Bell’s inequalities. • Squeezing entanglement by correlated homodyne measurements with variation of local oscillator phases.

47. II.D Squeezed Light & Continuous Variable Quantum Information

48. Analogue QIS • Continuous-Variable QI important as often the first realization of QI tasks, e.g. quantum teleportation, sharing quantum secrets. • Exploits squeezed state technology • Continuous-Variable QI relates to Qubit QI as analogue information relates to digital information. • Encoding QI via amplitude modulation. • Challenge for analogue information: error correction • What qualifies Analogue QI as quantum? (i) Exceed vacuum noise limit. (ii) Outperform analogue info processing.

49. Logic states • For qubit, logical basis: |0 and |1. • For CVQI, logical states are |x, for x real, but these states are not attainable physically. • Gaussian (squeezed) states approximate these logical states (but lose orthogonality ). • Coherent state (laser output) corresponds to a=1 • General state is

50. II.E Example of Quantum Optics in QI