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Basics of Quantum Measurement with Quantum Light

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Basics of Quantum Measurement with Quantum Light Basics of quantum measurement with quantum light Michael Hatridge University of Pittsburgh R. Vijayaraghavan TIFR Quantum light – Fock states Quantum state with definite number of photons in it! 푛 = 1,2,3 … n is an eigenstate of number operator 푛 = 푎†푎 푛 푛 = 푛 Let’s also look at what a coherent state looks like in terms of its 푎±푎† quadratures 퐼, 푄 = (equivalent to x and p for SHO) 2 Q I Quantum light – coherent states The quantum state of a driven harmonic oscillator ∞ 2 푛 − 훼 훼 훼 = 푒 2 푛! 푛=0 This state is not a eigenstate of photon number, but instead of lowering operator w/ complex eigenvalue α 푎 훼 = 훼 훼 In I/Q space, has gaussian distribution centered on coordinates 퐼, 푄 = (푅푒 훼 , 퐼푚(훼)) Q I Quantum light – squeezed states These basic quantum states have the same area in phase space Q Q I I We can consider other, more complicated states which ‘squeeze’ the light in certain quadratures while leaving area unchanged Q Q I I What to do with quantum light? • Detect/measure it to prove quantum properties - Q-function, Wigner Tomography - time/space correlations - bunching/anti-bunching • Use it as a tool/sensor - too many to count…. • Use it for quantum information - (flying) qubit - readout of another (stationary) qubit We must match the light to its detector • ‘Click’ or photon counting detectors project flying light onto a Fock state - great for Fock states, not a good basis for discriminating coherent light Output: ‘click’ or number w/ probability 2 푃푛 = 푛 휓푙ℎ푡 1 1 휓 = ( 훼 + 훼 ) 휓 = ( 푛 + 푚 ) 2 1 2 2 Q Q I vs. I Classical (microwave) amplifier In 퐺, 푁 Out Classical I/Q plot 푄 / 퐺 Q noise 표푢푡 noise grows! I 퐼표푢푡/ 퐺 signal signal unchanged - Amplifiers DEGRADE Signal-to-Noise Ratio (SNR) - Their job is to swamp noise of later elements Amplifiers have many properties First and foremost: Noise added by amplifier - Goes by many names: noise figure, noise factor, noise temperature - Here we’ll focus on how many noise quanta the amplifier adds. - In units of added quanta this amplifier adds 830 @ 7.5 GHz Quantum limit on phase-preserving amplifier Caves 1984 Vacuum state ( 0 ) Q Operators (I/Q) or (x,p) do not commute, and so there is minimum area in phase space, associated I w/ ½ photon energy. Mininum added noise is ½ photon 푄 / 퐺 Q 표푢푡 ½ photon ½ + ½ photon I 퐼표푢푡/ 퐺 signal unchanged Outline for the remainder of our tutorials Tutorial 1: Basics of quantum measurement with quantum light - Intro to parametric amplification - Classification of amplifiers by type and interaction - Limitations of parametric amplification - An introduction to superconducting circuits Tutorial 2: Quantum measurement of coherent states and qubits with phase- sensitive amplifiers - Phase-sensitive amplification with superconducting microwave circuits - Single-shot qubit readout - Weak measurements and quantum back-action - Entanglement via sequential measurement Tutorial 3: Quantum measurement of coherent states and qubits with phase- preserving amplifiers - Phase-preserving amplification with superconducting microwave circuits - Single-shot qubit readout - Weak measurements and quantum back-action - Entanglement via amplification Introduction to parametric amplifiers How to build a classical amplifier - Inductors, capacitors, resistors, transistors + DV voltage for gain How to build a quantum amplifier? - Inductors, capacitors, resistors, transistors + DV voltage for gain Quantum harmonic oscillators Nonlinear Hamiltonian + Ports parametric drive (no internal losses) Parametrically driven couplings direct exchange parametrically driven exchange (Conv) ℋ푛푡 ℋ = 휔 푎†푎 푛푡 = 휔 푏†푏 ℏ 푎 ℏ 푏 푎 푏 푎 푏 푐 ℋ푖푛푡 † † ℏ = 푔 푎푏 + 푎 푏 - if 휔 − 휔 ≫ 휅 this term dies ℋ푖푛푡 † † † 푎 푏 푎,푏 ℏ = 푔 푎푏 푐 + 푎 푏푐 due to energy conservation (RWA) - if 휔푐 ≠ 휔푎 − 휔푏 we can drive the c- mode ‘stiffly’ - interaction also turns off slowly vs. ℋ 푖푛푡 † ∗ † detuning, limiting the on/off ratio ℏ = 푔푎푏 + 푔 푎 푏 - Parametric drive fully controls strength and phase of interaction System Dynamics – Phase preserving amplification ℋ = 휔 푎†푎 + 휔 푏†푏 + 휔 푐†푐 + 푔 푎†푏†푐 + 푎푏푐† ℏ 푎 푏 푐 Pay careful attention to coupling, this form destroys one c ‘pump’ photon to create one photon each in a and b - Modes we use for quantum signals should be driven near their resonance frequency - We need the third ‘pump’ mode to be far away from the pump frequency so the pump can be ‘stiff’ and c becomes a number System Dynamics continued ℋ = 휔 푎†푎 + 휔 푏†푏 + 휔 푐†푐 + 푔 푎†푏†푐 + 푎푏푐† ℏ 푎 푏 푐 - We’ll see this explicitly later, but for now, consider the photon exchange process - If 휔푎 + 휔푏 = 휔푐 the process works, else suppressed by energy non-conservation - This feature will remain even after c-mode an (off-resonant) pump Master Eqn - In this example we will have our pump on, as well as a ‘weak’ signal in mode a - Mode b will be left ‘idle’ 푑푎 ℋ 휅 = [ , 푎] − 푎 푎 + 휅 푎푛 푑푡 ℏ 2 푎 푑푎 휅 = −휔 푎 − 푎 푎 − 푔푏†푐 + 휅 푎푛 푑푡 푎 2 푎 푑푏 휅 = −휔 푏 − 푏 푏 − 푔푎†푐 + 휅 푏푛 (푏푛= 0) 푑푡 푎 2 푏 Switch to Fourier Domain 푎(푡) → 푎 휔 Taking Fourier transform to make clear which frequencies are linked: - assuming we drive mode a at 휔1 = 휔푎 + Δ - signal in mode b will be at 휔2 = 휔푏 − Δ 휅 −휔 푎 휔 = −휔 푎 휔 − 푎 푎 휔 − 푔푏† 휔 푐 + 휅 푎푛 휔 1 1 푎 1 2 1 2 푎 1 휅 −휔 푏 휔 = −휔 푏 휔 − 푏 푏 휔 − 푔푎† 휔 푐 2 2 푏 2 2 2 1 where we have used the following relations: +∞ +∞ +∞ +∞ +∞ 푑푎 푑 휔1푡 휔1푡 휔1푡 휔1푡 휔 푡 푒 푑푡 = (푎푒 ) 푑푡 − 휔 푎푒 푑푡 = −휔 푎(휔 ) 푎 푡 푒 푑푡 = 푎[휔1] 푏 푡 푒 2 푑푡 = 푏[휔 ] 1 1 1 2 −∞ 푑푡 −∞ 푑푡 −∞ −∞ −∞ +∞ +∞ † 휔1푡 † − 휔1+휔2 푡 휔1푡 † 푐 = 푐 푒−휔푐푡 = 푐 푒−(휔1+휔2)푡 푔푏 푐푒 푑푡 = 푔푏 푐 푒 푒 푑푡 = 푔푏 (휔2) 푐 −∞ −∞ +∞ +∞ +∞ 푛 휔1푡 푛 −휔푡 +휔1푡 푛 푎 (푡)푒 푑푡 = 푑푡 푎 휔 푒 푒 푑휔 = 푎 휔1 −∞ −∞ −∞ Input-Output Theory Transmission lines 푎푛 푏표푢푡 푎 푏 푎표푢푡 푏푛 푏 + 푏 = κ 푏 푎푛 + 푎표푢푡 = κ푎푎 푛 표푢푡 푏 Substituting back into our equations for a and b we find: κ푎 1 + α 푔 푐 − Δ = − β + κ푎 2 κ푎 κ푏 κ β∗ 푔 푐 푏 + Δ = − (1 + α∗) 2 κ푏 κ푎 ϯ 푎표푢푡 푏 where α = and β = 표푢푡 푎푛 푎푛 Closed form solution 4 4 2 2 2 2 2 2 2 2 16푔 푐 + 8푔 푐 4Δ + κ푎κ푏 + 4Δ + κ푎 4Δ + κ푏 α = 4 4 2 2 2 2 2 2 2 16푔 푐 + 8푔 푐 4Δ − κ푎κ푏 + 4Δ + κ푎 4Δ + κ푏 20 For example : κ푎 = κ푏 = 50 푀퐻푧 10 푔 = 50 푀퐻푧 푐 2 = 0.36 Gain (dB) Gain -100 -50 0.0 50 100 Δ (MHz) 퐺 − 1푒휙푝 푎 퐺 퐺 푏 퐺 − 1푒−휙푝 Gain vs. pump power When Δ = 0, gain simplifies to: 2 2 2 2 1 + ρ 2 4푔 4 α = 2 where ρ = 푐 1 − ρ κ푎κ푏 100 Where ‘safe’ solution κ푎 = κ푏 = 50 푀퐻푧 50 푔 = 50 푀퐻푧 Gain (dB) Gain 0 0.5 1.0 푐 2 Amplifier bandwidth vs. gain and mode bandwidth 20 κ푎 = κ푏 = 70 푀퐻푧 κ푎 = κ푏 = 50 푀퐻푧 10 κ푎 = κ푏 = 20 푀퐻푧 Gain (dB) Gain 20 κ푎 = κ푏 = 50 푀퐻푧 퐺 10 2휋 퐵 ≃ κ푎κ푏 Gain (dB) Gain -50 0.0 50 Δ (MHz) Amplifier Limitations 퐺 − 1푒휙푝 1. It operates in reflection 푎 퐺 퐺 푏 퐺 − 1푒−휙푝 κ κ 2. It has narrow bandwidth 2휋 퐵 ≃ 푎 푏 퐺 Limitation 3: Gain Saturation G vs Psig unsaturated saturated 20 Gain (dB) 19 Pump depletion (PD) theory: • Psig , np , G • np , P-1dB P-1dB 18 -146 -138 -130 Signal Input Power (dBm) Problem: We don’t understand well all causes Pump depletion: Abdo Phys. Rev. B (2013); C. Eichler EPJ Quantum Technology (2014) But it is quantum limited! 퐺 − 1푒휙푝 푎 퐺 퐺 푏 퐺 − 1푒−휙푝 - Consider how much noise comes out of mode a with no inputs signals - The amplifier symmetrically handles both modes; the noise out of a is double its input due to the added noise from the ‘idler’ mode irrespective of what they are - For quantum signals you will get 1 photon total strong msmt 2000 fluctuations (1/2 + 1/2); second 1/2 physically sourced in idler mode - We’ll show how to test this with qubits in 휎 / 푚 1000 tutorial 3. 푄 counts 푔 푒 0 퐼푚/휎 Phase-sensitive amplifier ℋ푖푛푡 † † † ℏ = 푔 푎 푎 푐 + 푎푏푐 - Note that this is not the coupling Vijay will present tomorrow (his has 4 a operators; 1 each for signal and idler and 2 for pump) - To see the parallel, make the subsitution (a,b) - > (au , al ). An signal to au will be both amplified and coupled to al - Essentially one half of the amplifier acts as the ‘signal’ while the other half acts as the ‘idler’ Δ input input 휔푎 휔 휔푏 푎 퐺 − 1 퐺 퐺 − 1 퐺 Δ Δ output output 휔푎 휔푏 휔푎 Phase sensitive amplifier data - At zero offset frequencies these signal and idler tones interfere - One quadrature is amplified, the other deamplified - Which quadrature will be amplified is set by pump phase Signal input w/ Signal input here constant phase Freq @ 7.75027053 GHz ) ) -68 -70 dBm dBm Amplification -80 -70 -90 Deamplification -72 Δ푓 = 1 퐻푧 Integrated Power ( Integrated Power ( -100 7.74 7.75 7.76 0.0 0.5 1.0 Freq (GHz) Pump Phase / 2π Classifying amplifiers* 3-wave coupling 4-wave coupling one mode ‘Singly degenerate’ ‘Doubly degenerate’ phase-sensitive phase-sensitive two-mode ‘Non-degenerate’ ?? phase-preserving * At optical frequencies phase-preserving => heterodyne detection phase-sensitive => homodyne detection More useful: specify parametric interaction 3-wave coupling 4-wave coupling one mode 푔푎†푎†푐 푔푎†푎†푎푎 (a) two-mode 푔푎†푏†푐 푔푎†푏†푐푐 ∗∗ (a,b) ** I’m not aware of this one being built yet Other Couplings: Photon conversion 휔푝 = 휔푎 − 휔푏 ≠ 휔푐 † 휙푝 † −휙푝 퐻퐺 = ℏ푔 푎 푏푒 + 푎푏 푒 I 퐶푒+휙푝 푎 푏 푎 1 − 퐶 1 − 퐶 푏 퐶푒−휙푝 휔푝 ≃ 휔푎 − 휔푏 푃푃 푐 2 푃 conCorr=con+20.4 퐶 = 퐶 푃 2 09 1 + 푃 푃퐶 0 0 -15 -20 9.10 9.20 9.10 9.20 Frequency (GHz) Frequency (GHz) Sliwa PRX (2015) What about still odder couplings? Ex.: 푔푎†푎†푐푐푐 ? - This will be a phase-sensitive amplifier when driven at 2휔푎 = 3휔푐 - Such couplings may limit current device performance More exciting: multiple, simultaneous couplings 푄푏 - Can make non-reciprocal devices: circulators, directional amplifiers 훷푏 훷푎 푄푎 3-mode - Can also address other shortcomings coupler - Your theory proposal here! Bergeal et al.
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