10.6 Propagating quantum microwaves

Transcription

1 AS-Chap Propagating quantum microwaves

2 Propagating quantum microwaves emit Quantum - - Superconducting quantum circuits Artificial quantum matter Confined quantum states of light Does the emitted microwave radiation exhibit quantum properties? Commutation relations, superpositions, entanglement Quantum optics Yes, expected due to field quantization Confirmed by experiments Microwaves Expected in analogy to opics Different technology Experimental proof required! AS-Chap. 10-2

3 AS-Chap (Envisioned) applications of propagating quantum microwaves Quantum communication Quantum illumination φ Quantum information processing

4 AS-Chap Fundamental technological considerations Microwave losses may inhibit Observation of quantum properties of propagating microwaves Practical applications such as quantum microwave communication/illumination Superconducting cables Coherent propagation distance l coh sufficient? In resonators, microwave signals travel back and forth many times before losing coherence (T μs 1 ms) l coh m s T km comparable to optics Superconducting cables require cooling! Short- or medium-distance applications certainly feasible QIP platforms such as SQC also require cooling Compatible Technological compatibility to SQC No frequency conversion losses Natural candidate for chip-to-chip quantum communication between SQC

5 Fundamental technological considerations Microwave losses may inhibit Observation of quantum properties of propagating microwaves Practical applications such as quantum microwave communication/illumination Free-space propagation Atmospheric transparency windows Classical illumination with microwaves used for radar Known to pass through clouds, fog, and rain 20 GHz ta.edu/faculty/bau manpr/geosat2/rs -Introduction/RS- Introduction.html Typical frequencies 20 GHz (λ = 3 108m s 1.5 cm) f Compatible with SQC (superconducting gap of aluminum still twice as large) AS-Chap. 10-5

6 Quantization of the electromagnetic field F. D. Walls, G. Milburn, Quantum Optics (Springer, Berlin, 2008). Source-free (free field!) Maxwell equations B = 0, E = B D, D = 0, H = t t Coulomb gauge ( A = 0) B = A, E = A t (B = μ 0 H, D = ε 0 E, μ 0 ε 0 = c 2 ) A(r, t) satisfies wave equation 2 A = 1 2 A c 2 t 2 Separate vector potential A r, t = A + r, t + A (r, t) into Right-propagating components A + (r, t) varying with e iωt for ω > 0 Left-propagating components A (r, t) varying with e iωt for ω > 0 Restrict field to finite volume A + r, t = k c k u k r e iω kt Fourier coefficients c k constant for free field Vector mode functions u k r Satisfy wave equations 2 + ω k 2 u c 2 k r = 0 Satisfy transversality condition u k r = 0 Form orthonormal set V dr u k r u k r = δ kk Depend onboundary conditions AS-Chap. 10-6

7 Quantization of the electromagnetic field F. D. Walls, G. Milburn, Quantum Optics (Springer, Berlin, 2008). General examplef for boundary conditions Periodic (travelling waves) Reflecting walls (standing waves) Here: Plane wave functions suitable for cubic volume with side lengths L u k r = 1 e λ e ikr L 3/2 Typically no polarization in microwaves propagating in waveguides Polarization vector e λ = e Wave vector k = k x, k y, k z with k x,y,z = 2π n L x,y,z and n x,y,z Z e perpendicular to k Quantization of classical Fourier amplitudes a k, a k a k, a k with commutation relations a k, a k = δ kk A r, t = k ħ 2ω k ε 0 a k u k r e iωt + a k u k r e iωt E r, t = i k ħω k 2ε 0 a k u k r e iωt a k u k r e iωt Hamiltonian H = 1 2 Quantum states ψ k dr ε 0 E 2 + μ 0 H 2 = k ħω k a k a k of each mode can now be discussed independently! AS-Chap. 10-7

8 Continuos variables (CV) vs. discrete variables (DV) Classical single-mode electromagnetic waves A cos ωt + φ Equivalent description P cos ω k t + Q sin ω k t with field quadratures Q = A cos φ and P = A sin φ In engineering, P is often called I Field quadratures analogous to momentum/position in mechanics Annihilation operator a ω rc Φ+i q 2ω r Cħ Creation operator a ω rc Φ i q 2ω r Cħ H HO = E kin + E pot = p2 2m mω r 2 x 2 P 1 2ħmω r p 2, Q mω r 2ħ x2 Field quantization Q, P = i 2 ΔP ΔQ 1 4 Single-mode quantum field H HO = P 2 + Q 2 = ħω k a a Continuous-variable basis Set of eigenstates of either P or Q also forms basis Natural states are Gaussian states (coherent, squeezed, thermal) Discrete-variable basis Fock basis n Single photons 1 are the natural quantity of interest Any quantum state can be expressed either in CV or in DV Any quantum task (QIP, QSim, Qcomm, QIllu) can be expressed in CV or DV Nevertheless a particular basis may be more suitable for a particular problem AS-Chap. 10-8

9 AS-Chap Expressing quantum microwave states E. P. Menzel, PhD thesis (Tu München, 2013). General quantum state described by the desity matrix ρ = Ψ P Ψ Ψ Ψ P Ψ = Classical probability to be in state Ψ P Ψ > 0 and Ψ P Ψ = 1 Expectation value of operator O O = Tr O ρ Normalization Tr ρ = 1 Complex matrix entries Not easy to visualize Phase space representation of a quantum state Ideal classical states Points in phase space Noisy classical states Ordinary probability distribution P(q, p) in phase space q, p are the phase space variables associated with Q, P P q, p dqdp is the probability to find the system in state q, p Quantum states Heisenberg uncertainty relation ΔP ΔQ 1 as quantum noise 4 In general requires also negative probability densities Quasi-probability distribution W q, p with W q, p dqdp = 1 Wigner function W q, p = 1 ζ 2π q ρ q + ζ 2 2 eipζ dζ and ζ R

10 AS-Chap Wigner function examples E. P. Menzel, PhD thesis (Tu München, 2013). e 1 -contours n = 0 n = 1 Thermal state ρ th with n Tr a a ρ th = 1 W th q, p = 1 π n+ 1 2 e q2+p2 n+ 1 2 > 0 In the high-temperature limit k B T ħω k, thermal states can usually be considered to be classical Thermal states are aslo classical in the sense that they do not generate entanglement when applied to a beam splitter with vacuum in the other input

11 AS-Chap Wigner function examples E. P. Menzel, PhD thesis (Tu München, 2013). e 1 -contours α = 0 α = 4 Coherent state α Q + ip = D α 0 Produced by the displacement operator D α e α a α a applied to the vacuum W coh q, p = 2 π e 2 q Q 2 + p P 2 > 0 Coherent states are classical in the sense that the expectation values of the field operators obey the equations of motions for the actual field operators in the Heisenberg picture

12 Wigner function examples E. P. Menzel, PhD thesis (Tu München, 2013). e 1 -contours ξ = 0 ξ = i Squeezed vacuum determined by complex squeezing parameter ξ re iφ Produced by the squeezing operator S ξ e 1 2 ξ a 2 ξ a 2 applied to the vacuum r determines the amount of squeezing and φ the squeezing direction in phase space W sq q, p = 2 π e e2r +e 2r q+ip e2r e 2r e iφ q+ip 2 +e iφ q ip 2 > 0 Because ΔP 2 < 1, one must have ΔQ ΔP 2 to satsify the Heisenberg relation Squeezed states are nonclassical in the sense that they produce entanglement when applied to a beamsplitter with vacuum at the other input port AS-Chap

13 AS-Chap Wigner function examples 0 1 Number states Have a radially symmetric Wigner function The vacuum 0 is a Gaussian state with the vacuum variace of 0.5ħω_k Finite number states n 1 are nonclassical Because their Wigner function can become negative Because they when applied to a beamsplitter with vacuum at the other input port 5

14 State reconstruction of propagating quantum microwaves No efficient photon detectors Difficult task! Off-the-shelf linear amplifiers Add n 10 of noise to signal Two approaches: 1. Parametric amplifiers 2. Signal recovery methods (measure signal moments) Advanced microwave technology Realtime data processing FPGA WMI dual-path receiver extremedesignsonline.com AS-Chap

15 Dual-path state reconstruction of propagating quantum microwaves C 1 χ 1 χ 2 a + v 2 PC v BS C 2 a + v 2 Kowledge of all moments is equivalent to knowlege of the Wigner function or density matrix. Expectation values of all signal moment up to order n C 1 n 1 C 2 a n, χ 1 n, χ 2 n Iteratively obtain all signal & detector noise moments E.P. Menzel et al., PRL 105, (2010). a Nondeterministic & quantum signals up to order n Record all moments Q 1 j Q 2 k P 1 l P 2 m with j + k + l + m n and j, k, l, m N 0 AS-Chap

16 Dual-path state reconstruction of propagating quantum microwaves C 1 χ 1 χ 2 v a + v 2 PC BS C 2 a + v 2 Intuition for order n = 2 Use statistical independence of amplifier noise signals in different paths χ 1 χ 2 = χ 1 χ 2 = 0 C 1,2 = a + χ 1,2 = a + χ 1,2 = a C 1,2 2 = a + χ 1,2 2 = a 2 + aχ 1,2 + χ 1,2 a + a E.P. Menzel et al., PRL 105, (2010). AS-Chap