Strengthening weak measurements for qubit tomography and multitime correlators

Justin Dressel
Schmid College of Science and Technology

Institute for Quantum Studies

Chapman University

 

January 16, 2019

Justin Dressel  (PI)

 

José Raúl Gonzales Alonso (postdoc)

Razieh Mohseninia (postdoc)

Shiva Barzili  (grad student)

Lucas Burns (grad student, spring 2019)

Amy Lam (undergrad)

 


Luis Pedro García-Pintos  (postdoc - now at UMB)

Taylor Lee Patti (undergrad - now at Harvard)

William Parker (undergrad - now at U Oregon)

Aaron Grisez (undergrad - founder of Qhord.com)

Chapman Crew

Outline

  • Strengthening Weak Value Tomography
    • Weak Value Tutorial
    • Weak Value Tomography
    • Strong Weak Values with Neutrons

       
  • Strengthening Qubit Correlator Measurements
    • Weakly Measured Correlators
    • Generalized Qubit Measurements
    • Strongly Measured Correlators

What is a Weak Value?

Roughly, it is a conditioned expectation value under weak detector coupling that can have surprising properties

Aharonov, Albert, Vaidman (AAV), PRL 1988

What is a Conditioned Expectation Value?

A conditioned expectation value is the conditioned response of an indirectly coupled and properly calibrated meter

Pang, Dressel, Brun, PRL 113, 030401 (2014)

The detector response is encapsulated in the "measurement operator", which is a partial matrix element of the interaction propagator

Pang, Dressel, Brun, PRL 113, 030401 (2014)

What is a Conditioned Expectation Value?

What is a Weak Value?

Weak values appears as the key complex system parameters that characterize the detector response. Generally all infinite orders \((A^n)_w\) are needed, but for weaker coupling the first order suffices

Pang, Dressel, Brun, PRL 113, 030401 (2014)

Kofman, Ashab, Nori, Physics Reports 520, 43 (2012)

Dressel, Jordan, PRL 109, 230402 (2012)

Detector parameters:

System parameter:

Pang, Dressel, Brun, PRL 113, 030401 (2014)

What is a Weak Value?

Typically, only the linear response regime is considered, where the weak value is directly proportional to the detector response

What is a Weak Value?

Even keeping only the first-order weak value, the nonlinear expression is remarkably accurate even for moderately large coupling strengths

Example: Sagnac Interferometer

Prototype experiment: Howell lab, Rochester

PRL 102, 173601 (2009)

Ultra-sensitive to beam deflection: ~560 femto-radians of tilt detected

\phi\;\text{(phase offset)}
ϕ  (phase offset)\phi\;\text{(phase offset)}
k\;\text{(tilt)}
k  (tilt)k\;\text{(tilt)}
x\;\text{(signal)}
x  (signal)x\;\text{(signal)}
\sigma\;\text{(beam waist)}
σ  (beam waist)\sigma\;\text{(beam waist)}

Weak Value Analysis

|i\rangle \propto i|\!\circlearrowleft\rangle + |\!\circlearrowright\rangle
∣i⟩∝i∣​↺⟩+∣​↻⟩|i\rangle \propto i|\!\circlearrowleft\rangle + |\!\circlearrowright\rangle
|f\rangle \propto e^{i\phi/2}|\!\circlearrowleft\rangle - ie^{-i\phi/2}|\!\circlearrowright\rangle
∣f⟩∝eiϕ/2∣​↺⟩−ie−iϕ/2∣​↻⟩|f\rangle \propto e^{i\phi/2}|\!\circlearrowleft\rangle - ie^{-i\phi/2}|\!\circlearrowright\rangle
|\psi, i\rangle \to \langle p, f | \exp(-i k \hat{x}\hat{W}) | \psi, i \rangle \approx \langle f | i \rangle \langle p - k W_w | \psi \rangle
∣ψ,i⟩→⟨p,f∣exp⁡(−ikx^W^)∣ψ,i⟩≈⟨f∣i⟩⟨p−kWw∣ψ⟩|\psi, i\rangle \to \langle p, f | \exp(-i k \hat{x}\hat{W}) | \psi, i \rangle \approx \langle f | i \rangle \langle p - k W_w | \psi \rangle
\hat{W} = |\!\circlearrowleft\rangle\langle\!\circlearrowleft | - |\!\circlearrowright\rangle\langle\!\circlearrowright |
W^=∣​↺⟩⟨​↺∣−∣​↻⟩⟨​↻∣\hat{W} = |\!\circlearrowleft\rangle\langle\!\circlearrowleft | - |\!\circlearrowright\rangle\langle\!\circlearrowright |
W_w = \frac{\langle f |\hat{W}|i\rangle}{\langle f | i \rangle} = i\cot\frac{\phi}{2} \approx \frac{2i}{\phi}
Ww=⟨f∣W^∣i⟩⟨f∣i⟩=icot⁡ϕ2≈2iϕW_w = \frac{\langle f |\hat{W}|i\rangle}{\langle f | i \rangle} = i\cot\frac{\phi}{2} \approx \frac{2i}{\phi}
\phi\;\text{(phase offset)}
ϕ  (phase offset)\phi\;\text{(phase offset)}
k\;\text{(tilt)}
k  (tilt)k\;\text{(tilt)}
x\;\text{(signal)}
x  (signal)x\;\text{(signal)}
\sigma\;\text{(beam waist)}
σ  (beam waist)\sigma\;\text{(beam waist)}

Angular tilt (transverse momentum)  amplified by large weak value.

Weak value regime

k\sigma \ll \phi \ll 1
kσ≪ϕ≪1k\sigma \ll \phi \ll 1

Dark port has single lobe that approximates displaced a Gaussian centered at:

\langle x/\sigma \rangle \propto (\mathbf{k}/\phi)\sigma
⟨x/σ⟩∝(k/ϕ)σ\langle x/\sigma \rangle \propto (\mathbf{k}/\phi)\sigma

Tiny beam deflections can be distinguished, but with low output intensity.

I \propto \phi^2
I∝ϕ2I \propto \phi^2

Exact Collimated Analysis

|i\rangle = \frac{i|\!\circlearrowleft\rangle + |\!\circlearrowright\rangle}{\sqrt{2}}
∣i⟩=i∣​↺⟩+∣​↻⟩2|i\rangle = \frac{i|\!\circlearrowleft\rangle + |\!\circlearrowright\rangle}{\sqrt{2}}
|f\rangle = \frac{e^{i\phi/2}|\!\circlearrowleft\rangle - ie^{-i\phi/2}|\!\circlearrowright\rangle}{\sqrt{2}}
∣f⟩=eiϕ/2∣​↺⟩−ie−iϕ/2∣​↻⟩2|f\rangle = \frac{e^{i\phi/2}|\!\circlearrowleft\rangle - ie^{-i\phi/2}|\!\circlearrowright\rangle}{\sqrt{2}}
|\psi, i\rangle \to \langle x, f | \exp(-i k \hat{x}\hat{W}) | \psi, i \rangle = \langle x|\hat{M}_f\,|\psi\rangle = i\sin(\phi/2 - kx)\langle x | \psi\rangle
∣ψ,i⟩→⟨x,f∣exp⁡(−ikx^W^)∣ψ,i⟩=⟨x∣M^f ∣ψ⟩=isin⁡(ϕ/2−kx)⟨x∣ψ⟩|\psi, i\rangle \to \langle x, f | \exp(-i k \hat{x}\hat{W}) | \psi, i \rangle = \langle x|\hat{M}_f\,|\psi\rangle = i\sin(\phi/2 - kx)\langle x | \psi\rangle
\hat{W} = |\!\circlearrowleft\rangle\langle\!\circlearrowleft\! | - |\!\circlearrowright\rangle\langle\!\circlearrowright\! |
W^=∣​↺⟩⟨​↺​∣−∣​↻⟩⟨​↻​∣\hat{W} = |\!\circlearrowleft\rangle\langle\!\circlearrowleft\! | - |\!\circlearrowright\rangle\langle\!\circlearrowright\! |
\phi\;\text{(phase offset)}
ϕ  (phase offset)\phi\;\text{(phase offset)}
k\;\text{(tilt)}
k  (tilt)k\;\text{(tilt)}
x\;\text{(signal)}
x  (signal)x\;\text{(signal)}
\sigma\;\text{(beam waist)}
σ  (beam waist)\sigma\;\text{(beam waist)}
\hat{M}_f = \langle f | \exp(-i k \hat{x}\hat{W}) | i \rangle = i \sin(\phi/2 - k\hat{x})
M^f=⟨f∣exp⁡(−ikx^W^)∣i⟩=isin⁡(ϕ/2−kx^)\hat{M}_f = \langle f | \exp(-i k \hat{x}\hat{W}) | i \rangle = i \sin(\phi/2 - k\hat{x})
P(x) = |\langle x,f| \psi,i\rangle|^2 = \sin^2(\phi/2 - kx)\,P_0(x) = \frac{\sin^2(\phi/2 - kx)}{\sqrt{2\pi\sigma^2}} e^{-(x/\sigma)^2/2}
P(x)=∣⟨x,f∣ψ,i⟩∣2=sin⁡2(ϕ/2−kx) P0(x)=sin⁡2(ϕ/2−kx)2πσ2e−(x/σ)2/2P(x) = |\langle x,f| \psi,i\rangle|^2 = \sin^2(\phi/2 - kx)\,P_0(x) = \frac{\sin^2(\phi/2 - kx)}{\sqrt{2\pi\sigma^2}} e^{-(x/\sigma)^2/2}

Original profile of beam becomes modulated.

JD et al., PRA  88 , 023801 (2013)

Collimated Dark Port Profiles

Left: Wavefront tilt mechanism producing spatial modulation

Right : Asymmetric dark port profiles in different regimes

Dashed envelope: input beam intensity

Solid curve:

dark port intensity

 

Top right:

weak value regime
 

 

Middle right:

double lobe regime
 

 

Bottom right:

misaligned regime

k\sigma \ll \phi \ll 1
kσ≪ϕ≪1k\sigma \ll \phi \ll 1
\phi\ll k\sigma \ll 1
ϕ≪kσ≪1\phi\ll k\sigma \ll 1
\phi\ll 1 \ll k\sigma
ϕ≪1≪kσ\phi\ll 1 \ll k\sigma

Weak Value Tomography

\displaystyle |\psi\rangle = \alpha |H\rangle + \beta |V\rangle
∣ψ⟩=α∣H⟩+β∣V⟩\displaystyle |\psi\rangle = \alpha |H\rangle + \beta |V\rangle
\displaystyle c \equiv \frac{\langle D|H \rangle}{\langle D | \psi \rangle} = \frac{\langle D|V \rangle}{\langle D | \psi \rangle}
c≡⟨D∣H⟩⟨D∣ψ⟩=⟨D∣V⟩⟨D∣ψ⟩\displaystyle c \equiv \frac{\langle D|H \rangle}{\langle D | \psi \rangle} = \frac{\langle D|V \rangle}{\langle D | \psi \rangle}
\displaystyle \implies c|\psi\rangle = \frac{\langle D| H \rangle \langle H | \psi \rangle}{\langle D | \psi \rangle} |H\rangle + \frac{\langle D | V \rangle \langle V | \psi \rangle}{\langle D | \psi \rangle} |V\rangle
  ⟹  c∣ψ⟩=⟨D∣H⟩⟨H∣ψ⟩⟨D∣ψ⟩∣H⟩+⟨D∣V⟩⟨V∣ψ⟩⟨D∣ψ⟩∣V⟩\displaystyle \implies c|\psi\rangle = \frac{\langle D| H \rangle \langle H | \psi \rangle}{\langle D | \psi \rangle} |H\rangle + \frac{\langle D | V \rangle \langle V | \psi \rangle}{\langle D | \psi \rangle} |V\rangle

Measurable weak values

Determining these also determines the state, up to a simple renormalization

Unnormalized initial state:

Arbitrary constant, in terms of unbiased "postselection" basis

Unknown state:

Dressel, et al. RMP 86, 307 (2014)

Task: Measure the path state of a neutron inside a neutron interferometer using its coupled spin freedom as a probe 

Denkmayr, Dressel, et al. PRL 118, 010402 (2017)

  1. Spin-up neutrons preselected by silicon crystal Bragg condition of interferometer
  2. DC coil (ST1) rotates spin to +X  (initial spin state prepared at this point)
  3. Sapphire slab (PS) phase shifts relative paths (initial path state prepared at this point)
  4. Helmholtz coils (HCs) perform path-dependent spin rotations by tunable angle alpha
  5. After interferometer, DC coil (ST2) and CoTi supermirror (A) spin-polarizes neutrons
  6. Only neutrons at O-Det are analyzed (postselection completed here)
  7. Measuring average spin response allows determination of path weak values, and thus the intermediate path state inside the interferometer
|\Psi_i\rangle = (c_+ |P_I\rangle + c_- |P_{II}\rangle)\otimes |S_x; +\rangle
∣Ψi⟩=(c+∣PI⟩+c−∣PII⟩)⊗∣Sx;+⟩|\Psi_i\rangle = (c_+ |P_I\rangle + c_- |P_{II}\rangle)\otimes |S_x; +\rangle

Denkmayr, Dressel, et al. PRL 118, 010402 (2017)

\displaystyle |P_i\rangle = \frac{\langle\hat{\Pi}^p_{z+}\rangle_w |P_I\rangle + \langle\hat{\Pi}^p_{z-}\rangle_w |P_{II}\rangle }{\sqrt{|\langle\hat{\Pi}^p_{z+}\rangle_w |^2 + |\langle\hat{\Pi}^p_{z-}\rangle_w |^2}}
∣Pi⟩=⟨Π^z+p⟩w∣PI⟩+⟨Π^z−p⟩w∣PII⟩∣⟨Π^z+p⟩w∣2+∣⟨Π^z−p⟩w∣2\displaystyle |P_i\rangle = \frac{\langle\hat{\Pi}^p_{z+}\rangle_w |P_I\rangle + \langle\hat{\Pi}^p_{z-}\rangle_w |P_{II}\rangle }{\sqrt{|\langle\hat{\Pi}^p_{z+}\rangle_w |^2 + |\langle\hat{\Pi}^p_{z-}\rangle_w |^2}}
\displaystyle \langle\hat{\Pi}^p_{z\pm}\rangle_w = \frac{1 \pm \langle \hat{\sigma}_z^p\rangle_w}{2}
⟨Π^z±p⟩w=1±⟨σ^zp⟩w2\displaystyle \langle\hat{\Pi}^p_{z\pm}\rangle_w = \frac{1 \pm \langle \hat{\sigma}_z^p\rangle_w}{2}

Expressions exact for any relative spin-splitting angle alpha

Usual linear response regime when alpha small

Angle alpha selected here

Spin measurement axis selected here

Arbitrary-strength Path Weak Values

Relative phase of path state varied, keeping relative magnitudes the same (only imaginary weak value nonzero)

Weak (left) and Strong (right) measurements compared

Different-looking responses yield same weak value reconstructions

Denkmayr, Dressel, et al. Physica B (2018)

Same Weak Values : Stronger has better accuracy and precision

Denkmayr, Dressel, et al. Physica B (2018)

Outline

  • Strengthening Weak Value Tomography
    • Weak Value Tutorial
    • Weak Value Tomography
    • Strong Weak Values with Neutrons

       
  • Strengthening Qubit Correlator Measurements
    • Weakly Measured Correlators
    • Generalized Qubit Measurements
    • Strongly Measured Correlators

Recall Weak Measurement

What happens when you make several weak measurements in a row?

Pang, Dressel, Brun, PRL 113, 030401 (2014)

\hat{M} = \langle \psi_f | \exp(-i (g/2) \hat{A}\otimes \hat{D} ) |\psi_i \rangle \approx \sqrt{p}\,\hat{1} + (g/2)\,\hat{A} + O(g^2)
M^=⟨ψf∣exp⁡(−i(g/2)A^⊗D^)∣ψi⟩≈p 1^+(g/2) A^+O(g2)\hat{M} = \langle \psi_f | \exp(-i (g/2) \hat{A}\otimes \hat{D} ) |\psi_i \rangle \approx \sqrt{p}\,\hat{1} + (g/2)\,\hat{A} + O(g^2)

Crude idea : measurement operator approximately linearizes in weak measurement regime (linear response regime)

Post-interaction system state couples to anticommutator to linear order in coupling:

\displaystyle \text{Tr}(\hat{M}\hat{\rho}\hat{M}^\dagger) \approx p + g\sqrt{p}\,\text{Tr}\left[\frac{\hat{A}\hat{\rho}+\hat{\rho}\hat{A}}{2}\right] + O(g^2)
Tr(M^ρ^M^†)≈p+gp Tr[A^ρ^+ρ^A^2]+O(g2)\displaystyle \text{Tr}(\hat{M}\hat{\rho}\hat{M}^\dagger) \approx p + g\sqrt{p}\,\text{Tr}\left[\frac{\hat{A}\hat{\rho}+\hat{\rho}\hat{A}}{2}\right] + O(g^2)

Sequences of measurements create nested anticommutator terms to lowest order in the joint couplings

\displaystyle \text{Tr}(\hat{N}\hat{M}\hat{\rho}\hat{M}^\dagger\hat{N}^\dagger) \approx p_1p_2 + g_1\sqrt{p_1}p_2\,\text{Tr}\left[\frac{\{\hat{A}_1,\hat{\rho}\}}{2}\right] + g_2\sqrt{p_2}p_1\,\text{Tr}\left[\frac{\{\hat{A}_2,\hat{\rho}\}}{2}\right] + g_1g_2\sqrt{p_1p_2}\,\text{Tr}\left[\frac{\{\hat{A}_2,\{\hat{A}_1,\hat{\rho}\}\}}{4}\right] + \cdots
Tr(N^M^ρ^M^†N^†)≈p1p2+g1p1p2 Tr[{A^1,ρ^}2]+g2p2p1 Tr[{A^2,ρ^}2]+g1g2p1p2 Tr[{A^2,{A^1,ρ^}}4]+⋯\displaystyle \text{Tr}(\hat{N}\hat{M}\hat{\rho}\hat{M}^\dagger\hat{N}^\dagger) \approx p_1p_2 + g_1\sqrt{p_1}p_2\,\text{Tr}\left[\frac{\{\hat{A}_1,\hat{\rho}\}}{2}\right] + g_2\sqrt{p_2}p_1\,\text{Tr}\left[\frac{\{\hat{A}_2,\hat{\rho}\}}{2}\right] + g_1g_2\sqrt{p_1p_2}\,\text{Tr}\left[\frac{\{\hat{A}_2,\{\hat{A}_1,\hat{\rho}\}\}}{4}\right] + \cdots
\displaystyle \text{Tr}(\hat{N}\hat{M}\hat{\rho}\hat{M}^\dagger\hat{N}^\dagger) \approx p_1p_2 + g_1\sqrt{p_1}p_2\,\text{Tr}\left[\frac{\{\hat{A}_1,\hat{\rho}\}}{2}\right] + g_2\sqrt{p_2}p_1\,\text{Tr}\left[\frac{\{\hat{A}_2,\hat{\rho}\}}{2}\right] + g_1g_2\sqrt{p_1p_2}\,\text{Tr}\left[\frac{\{\hat{A}_2,\{\hat{A}_1,\hat{\rho}\}\}}{4}\right] + \cdots
Tr(N^M^ρ^M^†N^†)≈p1p2+g1p1p2 Tr[{A^1,ρ^}2]+g2p2p1 Tr[{A^2,ρ^}2]+g1g2p1p2 Tr[{A^2,{A^1,ρ^}}4]+⋯\displaystyle \text{Tr}(\hat{N}\hat{M}\hat{\rho}\hat{M}^\dagger\hat{N}^\dagger) \approx p_1p_2 + g_1\sqrt{p_1}p_2\,\text{Tr}\left[\frac{\{\hat{A}_1,\hat{\rho}\}}{2}\right] + g_2\sqrt{p_2}p_1\,\text{Tr}\left[\frac{\{\hat{A}_2,\hat{\rho}\}}{2}\right] + g_1g_2\sqrt{p_1p_2}\,\text{Tr}\left[\frac{\{\hat{A}_2,\{\hat{A}_1,\hat{\rho}\}\}}{4}\right] + \cdots
  • Benefit : Isolating such a nested commutator term gives access to interesting temporal correlation information that is sensitive to coherence (e.g., quasiprobability characterizations of the dynamics : Yunger Halpern, Swingle, Dressel PRA 9, 042105 (2018))
     
  • Problem : Isolating this term means subtracting away unwanted terms, including the higher-order coupling terms, which is difficult in practice with realistic measurements
     
  • Miracle : For qubits, this term can be isolated exactly with any measurement strengths g  (Dressel, et al. PRA 98, 012032 (2018) )

Arbitrary Strength Qubit Measurements

Informative Measurement:

Non-Informative Measurement:

\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
M^ϕ,a(A)=12[cos⁡ϕ21^−(−1)asin⁡ϕ2A^]\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
\displaystyle \hat{N}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} +i(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
N^ϕ,a(A)=12[cos⁡ϕ21^+i(−1)asin⁡ϕ2A^]\displaystyle \hat{N}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} +i(-1)^a \sin\frac{\phi}{2}\hat{A}\right]

Dressel, et al. PRA 98, 012032 (2018)

\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{\dagger(A)}\hat{M}_{\phi,a}^{(A)} = \hat{A}
∑a=0,1αϕ,a M^ϕ,a†(A)M^ϕ,a(A)=A^\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{\dagger(A)}\hat{M}_{\phi,a}^{(A)} = \hat{A}
\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
αϕ,a=(−1)a+1sin⁡ϕ\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}

Generalized spectral decomposition

(Works for observables A that square to the identity)

Arbitrary Strength Qubit Measurements

Informative Measurement:

White, Mutus, Dressel, et al. npj Quantum Information 2, 15022 (2016)

\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
M^ϕ,a(A)=12[cos⁡ϕ21^−(−1)asin⁡ϕ2A^]\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{\dagger(A)}\hat{M}_{\phi,a}^{(A)} = \hat{A}
∑a=0,1αϕ,a M^ϕ,a†(A)M^ϕ,a(A)=A^\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{\dagger(A)}\hat{M}_{\phi,a}^{(A)} = \hat{A}
\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
αϕ,a=(−1)a+1sin⁡ϕ\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}

Generalized spectral decomposition

Generalized measurement method already used experimentally with superconducting qubits (Google)

Arbitrary Strength Qubit Measurements

Informative Measurement:

White, Mutus, Dressel, et al. npj Quantum Information 2, 15022 (2016)

Arbitrary Strength Qubit Measurements

Informative Measurement:

\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
M^ϕ,a(A)=12[cos⁡ϕ21^−(−1)asin⁡ϕ2A^]\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]

Dressel, et al. PRA 98, 012032 (2018)

Useful corollaries:

\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{\dagger(A)}\hat{M}_{\phi,a}^{(A)} = \hat{A}
∑a=0,1αϕ,a M^ϕ,a†(A)M^ϕ,a(A)=A^\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{\dagger(A)}\hat{M}_{\phi,a}^{(A)} = \hat{A}
\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
αϕ,a=(−1)a+1sin⁡ϕ\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{(A)}\hat{\rho}\hat{M}_{\phi,a}^{\dagger(A)} = \frac{\{\hat{A},\hat{\rho}\}}{2}
∑a=0,1αϕ,a M^ϕ,a(A)ρ^M^ϕ,a†(A)={A^,ρ^}2\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{(A)}\hat{\rho}\hat{M}_{\phi,a}^{\dagger(A)} = \frac{\{\hat{A},\hat{\rho}\}}{2}

Anticommutators isolated perfectly for any coupling strength

\displaystyle \sum_{a_1,\ldots,a_m=0,1} \alpha_{\phi_1,a_1}\cdots\alpha_{\phi_m,a_m}\,P(a_1,\ldots,a_m) = \left\langle\frac{\{\cdots\{\{\hat{A}_m,\hat{A}_{m-1}\},\hat{A}_{m-2}\}\cdots,\hat{A}_1\}}{2^{m-1}}\right\rangle_\rho
∑a1,…,am=0,1αϕ1,a1⋯αϕm,am P(a1,…,am)=⟨{⋯{{A^m,A^m−1},A^m−2}⋯ ,A^1}2m−1⟩ρ\displaystyle \sum_{a_1,\ldots,a_m=0,1} \alpha_{\phi_1,a_1}\cdots\alpha_{\phi_m,a_m}\,P(a_1,\ldots,a_m) = \left\langle\frac{\{\cdots\{\{\hat{A}_m,\hat{A}_{m-1}\},\hat{A}_{m-2}\}\cdots,\hat{A}_1\}}{2^{m-1}}\right\rangle_\rho

Nested anticommutator averages can be measured directly for any coupling strength

Arbitrary Strength Qubit Measurements

Non-Informative Measurement:

Dressel, et al. PRA 98, 012032 (2018)

Useful corollaries:

\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
αϕ,a=(−1)a+1sin⁡ϕ\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{N}_{\phi,a}^{(A)}\hat{\rho}\hat{N}_{\phi,a}^{\dagger(A)} = \frac{[\hat{A},\hat{\rho}]}{2i}
∑a=0,1αϕ,a N^ϕ,a(A)ρ^N^ϕ,a†(A)=[A^,ρ^]2i\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{N}_{\phi,a}^{(A)}\hat{\rho}\hat{N}_{\phi,a}^{\dagger(A)} = \frac{[\hat{A},\hat{\rho}]}{2i}

Commutators also isolated perfectly for any coupling strength

\displaystyle \sum_{\tilde{a}_1,\ldots,a_m=0,1} \alpha_{\phi_1,\tilde{a}_1}\cdots\alpha_{\phi_m,a_m}\,P(\tilde{a}_1,\ldots,a_m) = \left\langle\frac{[\cdots\{\{\hat{A}_m,\hat{A}_{m-1}\},\hat{A}_{m-2}\}\cdots,\hat{A}_1]}{2^{m-2}(2i)}\right\rangle_\rho
∑a~1,…,am=0,1αϕ1,a~1⋯αϕm,am P(a~1,…,am)=⟨[⋯{{A^m,A^m−1},A^m−2}⋯ ,A^1]2m−2(2i)⟩ρ\displaystyle \sum_{\tilde{a}_1,\ldots,a_m=0,1} \alpha_{\phi_1,\tilde{a}_1}\cdots\alpha_{\phi_m,a_m}\,P(\tilde{a}_1,\ldots,a_m) = \left\langle\frac{[\cdots\{\{\hat{A}_m,\hat{A}_{m-1}\},\hat{A}_{m-2}\}\cdots,\hat{A}_1]}{2^{m-2}(2i)}\right\rangle_\rho

Nested commutator averages can also be measured directly for any coupling strength

\displaystyle \hat{N}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} +i(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
N^ϕ,a(A)=12[cos⁡ϕ21^+i(−1)asin⁡ϕ2A^]\displaystyle \hat{N}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} +i(-1)^a \sin\frac{\phi}{2}\hat{A}\right]

Arbitrary Strength Qubit Measurements

Dressel, et al. PRA 98, 012032 (2018)

\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
αϕ,a=(−1)a+1sin⁡ϕ\displaystyle \alpha_{\phi,a} = \frac{(-1)^{a+1}}{\sin\phi}
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{N}_{\phi,a}^{(A)}\hat{\rho}\hat{N}_{\phi,a}^{\dagger(A)} = \frac{[\hat{A},\hat{\rho}]}{2i}
∑a=0,1αϕ,a N^ϕ,a(A)ρ^N^ϕ,a†(A)=[A^,ρ^]2i\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{N}_{\phi,a}^{(A)}\hat{\rho}\hat{N}_{\phi,a}^{\dagger(A)} = \frac{[\hat{A},\hat{\rho}]}{2i}
\displaystyle \hat{N}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} +i(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
N^ϕ,a(A)=12[cos⁡ϕ21^+i(−1)asin⁡ϕ2A^]\displaystyle \hat{N}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} +i(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
M^ϕ,a(A)=12[cos⁡ϕ21^−(−1)asin⁡ϕ2A^]\displaystyle \hat{M}_{\phi,a}^{(A)} = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^a \sin\frac{\phi}{2}\hat{A}\right]
\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{(A)}\hat{\rho}\hat{M}_{\phi,a}^{\dagger(A)} = \frac{\{\hat{A},\hat{\rho}\}}{2}
∑a=0,1αϕ,a M^ϕ,a(A)ρ^M^ϕ,a†(A)={A^,ρ^}2\displaystyle \sum_{a=0,1} \alpha_{\phi,a}\,\hat{M}_{\phi,a}^{(A)}\hat{\rho}\hat{M}_{\phi,a}^{\dagger(A)} = \frac{\{\hat{A},\hat{\rho}\}}{2}

Why does this work? What about state collapse?

(Lindblad) decoherence terms from the measurement cancel in the average

Any marginalization of results exposes the collapse-based disturbance

Strongly Measured 2-Point Correlator

Dressel, et al. PRA 98, 012032 (2018)

\displaystyle \hat{U}_t^\dagger\hat{M}_{\phi,b}^{(B)}\hat{U}_t = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^b \sin\frac{\phi}{2}\hat{B}(t)\right]
U^t†M^ϕ,b(B)U^t=12[cos⁡ϕ21^−(−1)bsin⁡ϕ2B^(t)]\displaystyle \hat{U}_t^\dagger\hat{M}_{\phi,b}^{(B)}\hat{U}_t = \frac{1}{\sqrt{2}}\left[ \cos\frac{\phi}{2}\hat{1} -(-1)^b \sin\frac{\phi}{2}\hat{B}(t)\right]

2-point temporal qubit correlators may be directly measured using any measurement strength

These are common theoretical objects that are usually inaccessible in measurements directly

Bracketing measurement by unitary evolutions probes a Heisenberg-evolved operator

The final unitary may be omitted as inconsequential to the measured average

\displaystyle \sum_{a,b=0,1} \alpha_{\phi_a,a}\cdots\alpha_{\phi_b,b}\,P(a,b) = \left\langle\frac{\{\hat{B}(t),\hat{A}\}}{2}\right\rangle_\rho = \text{Re}\langle\hat{B}(t)\hat{A}\rangle_\rho
∑a,b=0,1αϕa,a⋯αϕb,b P(a,b)=⟨{B^(t),A^}2⟩ρ=Re⟨B^(t)A^⟩ρ\displaystyle \sum_{a,b=0,1} \alpha_{\phi_a,a}\cdots\alpha_{\phi_b,b}\,P(a,b) = \left\langle\frac{\{\hat{B}(t),\hat{A}\}}{2}\right\rangle_\rho = \text{Re}\langle\hat{B}(t)\hat{A}\rangle_\rho
\displaystyle \sum_{a,b=0,1} \alpha_{\phi_a,\tilde{a}}\cdots\alpha_{\phi_b,b}\,P(\tilde{a},b) = \left\langle\frac{[\hat{B}(t),\hat{A}]}{2i}\right\rangle_\rho = \text{Im}\langle\hat{B}(t)\hat{A}\rangle_\rho
∑a,b=0,1αϕa,a~⋯αϕb,b P(a~,b)=⟨[B^(t),A^]2i⟩ρ=Im⟨B^(t)A^⟩ρ\displaystyle \sum_{a,b=0,1} \alpha_{\phi_a,\tilde{a}}\cdots\alpha_{\phi_b,b}\,P(\tilde{a},b) = \left\langle\frac{[\hat{B}(t),\hat{A}]}{2i}\right\rangle_\rho = \text{Im}\langle\hat{B}(t)\hat{A}\rangle_\rho

Strongly Measured 4-Point Correlator

Dressel, et al. PRA 98, 012032 (2018)

4-point temporal qubit out-of-time-ordered correlators may be directly measured using any measurement strength

These are recently explored theoretical objects connected with information scrambling and chaos

\displaystyle \sum_{a,b,a',b'=0,1} \alpha_{\phi_a,a}\alpha_{\phi_b,b}\alpha_{\phi_{a'},a'}\alpha_{\phi_{b'},b'}\,P(a,b,a',b') = \left\langle\frac{\{\{\{\hat{B}(t),\hat{A}\},\hat{B}(t)\},\hat{A}\}}{2^3}\right\rangle_\rho = \frac{1 + \text{Re}\langle\hat{B}(t)\hat{A}\hat{B}(t)\hat{A}\rangle_\rho}{2}
∑a,b,a′,b′=0,1αϕa,aαϕb,bαϕa′,a′αϕb′,b′ P(a,b,a′,b′)=⟨{{{B^(t),A^},B^(t)},A^}23⟩ρ=1+Re⟨B^(t)A^B^(t)A^⟩ρ2\displaystyle \sum_{a,b,a',b'=0,1} \alpha_{\phi_a,a}\alpha_{\phi_b,b}\alpha_{\phi_{a'},a'}\alpha_{\phi_{b'},b'}\,P(a,b,a',b') = \left\langle\frac{\{\{\{\hat{B}(t),\hat{A}\},\hat{B}(t)\},\hat{A}\}}{2^3}\right\rangle_\rho = \frac{1 + \text{Re}\langle\hat{B}(t)\hat{A}\hat{B}(t)\hat{A}\rangle_\rho}{2}
\displaystyle \sum_{\tilde{a},b,a',b'=0,1} \alpha_{\phi_a,\tilde{a}}\alpha_{\phi_b,b}\alpha_{\phi_{a'},a'}\alpha_{\phi_{b'},b'}\,P(\tilde{a},b,a',b') = \left\langle\frac{[\{\{\hat{B}(t),\hat{A}\},\hat{B}(t)\},\hat{A}]}{2^2(2i)}\right\rangle_\rho = \frac{\text{Im}\langle\hat{B}(t)\hat{A}\hat{B}(t)\hat{A}\rangle_\rho}{2}
∑a~,b,a′,b′=0,1αϕa,a~αϕb,bαϕa′,a′αϕb′,b′ P(a~,b,a′,b′)=⟨[{{B^(t),A^},B^(t)},A^]22(2i)⟩ρ=Im⟨B^(t)A^B^(t)A^⟩ρ2\displaystyle \sum_{\tilde{a},b,a',b'=0,1} \alpha_{\phi_a,\tilde{a}}\alpha_{\phi_b,b}\alpha_{\phi_{a'},a'}\alpha_{\phi_{b'},b'}\,P(\tilde{a},b,a',b') = \left\langle\frac{[\{\{\hat{B}(t),\hat{A}\},\hat{B}(t)\},\hat{A}]}{2^2(2i)}\right\rangle_\rho = \frac{\text{Im}\langle\hat{B}(t)\hat{A}\hat{B}(t)\hat{A}\rangle_\rho}{2}

(one reversed evolution needed)

Conclusions

  • Weak values are better when made strong
  • Nested (anti)commutators are accessible for qubits at any strength
  • 2- and 4-point qubit correlators can be directly measured
     
  • Follow-up research in the works:
    • Measuring qubit quasiprobabilities strongly with multiple setups
    • Measuring qutrit correlators with arbitrary strength
    • Investigation of out-of-time-ordered correlators experimentally

Thank you!

Superconducting Qubits:

Review and recent results

Superconducting Qubits

\hbar\omega_q
ℏωq\hbar\omega_q
\hbar(\omega_q - \delta_q)
ℏ(ωq−δq)\hbar(\omega_q - \delta_q)
\hbar(\omega_q - 2\delta_q)
ℏ(ωq−2δq)\hbar(\omega_q - 2\delta_q)

Mesoscopic quantum coherence of collective charge motion at \(\mu\)m scale

EM Fields produced by charge motion described by Circuit QED

Lowest levels of anharmonic oscillator potentials treated as artificial atoms

Typical Transmon Parameters

\displaystyle \hat{H} = \hbar\hat{\Omega} = \frac{\hbar\omega_q}{2}\hat{\sigma}_z
H^=ℏΩ^=ℏωq2σ^z\displaystyle \hat{H} = \hbar\hat{\Omega} = \frac{\hbar\omega_q}{2}\hat{\sigma}_z
\hbar\omega_q
ℏωq\hbar\omega_q
\hbar(\omega_q - \delta_q)
ℏ(ωq−δq)\hbar(\omega_q - \delta_q)
\hbar(\omega_q - 2\delta_q)
ℏ(ωq−2δq)\hbar(\omega_q - 2\delta_q)
\omega_q/2\pi \sim \text{4--7GHz}
ωq/2π∼4–7GHz\omega_q/2\pi \sim \text{4--7GHz}
\delta_q/2\pi \sim \text{100--300MHz}
δq/2π∼100–300MHz\delta_q/2\pi \sim \text{100--300MHz}

Microwave Measurement

Note : Without a quantum-limited amplifier, this doesn't work!

 

The Josephson Parametric Amplifier (JPA) and Traveling Wave Parametric Amplifier (TWPA) boost signal enough for later HEMTs in the readout chain to resolve the information.

"3D Transmon"

\hbar\omega_q
ℏωq\hbar\omega_q
\hbar(\omega_q - \delta_q)
ℏ(ωq−δq)\hbar(\omega_q - \delta_q)
\hbar(\omega_q - 2\delta_q)
ℏ(ωq−2δq)\hbar(\omega_q - 2\delta_q)
\displaystyle \hat{H}_r = \hbar\omega_r\hat{a}^\dagger\hat{a}
H^r=ℏωra^†a^\displaystyle \hat{H}_r = \hbar\omega_r\hat{a}^\dagger\hat{a}

Cavity mode:

\omega_r/2\pi \sim \text{5--7GHz}
ωr/2π∼5–7GHz\omega_r/2\pi \sim \text{5--7GHz}
|\Delta|\equiv|\omega_q-\omega_r| \gg g
∣Δ∣≡∣ωq−ωr∣≫g|\Delta|\equiv|\omega_q-\omega_r| \gg g
\displaystyle \hat{H}_{c} = \hbar g(\hat{a}+\hat{a}^\dagger)\hat{\sigma}_x
H^c=ℏg(a^+a^†)σ^x\displaystyle \hat{H}_{c} = \hbar g(\hat{a}+\hat{a}^\dagger)\hat{\sigma}_x
g/2\pi \sim \text{100MHz}
g/2π∼100MHzg/2\pi \sim \text{100MHz}

Detuned (dispersive) regime (RWA):

\displaystyle \hat{H}_c \sim \hbar \chi\,\hat{a}^\dagger\hat{a}\,\hat{\sigma}_z
H^c∼ℏχ a^†a^ σ^z\displaystyle \hat{H}_c \sim \hbar \chi\,\hat{a}^\dagger\hat{a}\,\hat{\sigma}_z

X-X Coupling:

\displaystyle \chi \sim \left[\frac{g}{\Delta}\right]^2\!\delta_q \sim 1\text{MHz}
χ∼[gΔ]2​δq∼1MHz\displaystyle \chi \sim \left[\frac{g}{\Delta}\right]^2\!\delta_q \sim 1\text{MHz}
\displaystyle {\small \chi \neq \frac{g^2}{|\Delta|}}
χ≠g2∣Δ∣\displaystyle {\small \chi \neq \frac{g^2}{|\Delta|}}

Korotkov group, Phys. Rev. A 92, 012325 (2015)

Martinis group, Phys. Rev. Lett. 117, 190503 (2016)

2D planar chip schematic

Bus acts as Purcell filter, coupled to traveling wave parametric amplifier (TWPA)

\omega_r/2\pi \sim \text{5--7GHz}
ωr/2π∼5–7GHz\omega_r/2\pi \sim \text{5--7GHz}
g/2\pi \sim \text{100MHz}
g/2π∼100MHzg/2\pi \sim \text{100MHz}
\displaystyle \chi \sim \left[\frac{g}{\Delta}\right]^2\!\delta_q \sim 1\text{MHz}
χ∼[gΔ]2​δq∼1MHz\displaystyle \chi \sim \left[\frac{g}{\Delta}\right]^2\!\delta_q \sim 1\text{MHz}
\omega_q/2\pi \sim \text{4--7GHz}
ωq/2π∼4–7GHz\omega_q/2\pi \sim \text{4--7GHz}
\delta_q/2\pi \sim \text{100--300MHz}
δq/2π∼100–300MHz\delta_q/2\pi \sim \text{100--300MHz}

Similar parameters:

v1

v3

Coming soon: two-layer design of 20+ qubits
separated from control circuitry
(similar to Google Bristlecone + IBM Q)

Now on v8+

T_2 \sim 100\mu\text{s}
T2∼100μsT_2 \sim 100\mu\text{s}
T_1 \sim 60\mu\text{s}
T1∼60μsT_1 \sim 60\mu\text{s}

UCB 2D planar chip

Multiplexed 10 qubit control and readout

Single shot "projective" readout :
typical quantum computing goal

Microwave Measurement

A Brief Tour of Recent Results

Individual quantum state trajectories filtered from the readout are verified via spot checking predicted subensembles with tomography

Approach:

  • Perform a random tomographic pulse at the end of each data run
  • Spot check subensembles of data to verify tomography of final state

Quantum Trajectories

Murch et al., Nature 502, 211 (2013)

Monitored Rabi Drive

 JD and Siddiqi Group, Nature 511, 570 (2014)

Partial collapses compete with unitary dynamics

 

Ensemble-averaging the stochastic evolution recovers the usual Lindblad dynamics

Conditioned State Dynamics

 JD and Siddiqi Group, Nature 511, 570 (2014)

Experimental most probable path matches ODE solution derived from stochastic path integral

 JD and Jordan Group, PRA 88, 042110 (2013)

Tracking Drifting Parameters

Maximum likelihood techniques allow extraction of parameters drifts from stochastic records with reasonable precision

 JD and Jordan Group, PRA 95, 012314 (2017)

Feedback State Stabilization

Linear feedback (with very small temporal delays) can stabilize the qubit state to targeted regions of the Bloch sphere.

 JD and Jordan Group, PRA 96, 022311 (2017)

  • Symmetrically detuned pumps
     
  • Beats stroboscopically measure rotating qubit
     
  • Yields displacement coupling
     
  • Allows tunable measurement axis
     
  • Multiple cavity modes = multiple observables 

    •  
\hat{H}^{\text{rf}}_{\text{q-r}} = \chi\,(\hat{a}+\hat{a}^\dagger)\,\hat{\sigma}_{\delta}
H^q-rrf=χ (a^+a^†) σ^δ\hat{H}^{\text{rf}}_{\text{q-r}} = \chi\,(\hat{a}+\hat{a}^\dagger)\,\hat{\sigma}_{\delta}

Displacement coupling:

\hat{\sigma}_{\delta} = \cos\delta\,\hat{\sigma}^{\text{rf}}_x + \sin\delta\,\hat{\sigma}^{\text{rf}}_y
σ^δ=cos⁡δ σ^xrf+sin⁡δ σ^yrf\hat{\sigma}_{\delta} = \cos\delta\,\hat{\sigma}^{\text{rf}}_x + \sin\delta\,\hat{\sigma}^{\text{rf}}_y

Siddiqi group, Nature 538, 491 (2016)

Stroboscopic Displacement Coupling

\hat{H}_{\text{q}} = \frac{\hbar\Omega_R}{2}\hat{\sigma}_x
H^q=ℏΩR2σ^x\hat{H}_{\text{q}} = \frac{\hbar\Omega_R}{2}\hat{\sigma}_x

Rotating frame:

Incoherent Zeno-Dragging Qubit Gate

Idea : use time-varying measurement axes to drag the quantum state around the Bloch sphere using the quantum Zeno effect 


The record tracks the state well in this regime, so can be used as a herald for high-fidelity gates

Non-unitary gate

(measurement-based)

\hat{H}_{\text{q-r}}(t) = \chi\,(\hat{a}+\hat{a}^\dagger)\,\hat{\sigma}_{\delta(t)}
H^q-r(t)=χ (a^+a^†) σ^δ(t)\hat{H}_{\text{q-r}}(t) = \chi\,(\hat{a}+\hat{a}^\dagger)\,\hat{\sigma}_{\delta(t)}
\hat{\sigma}_{\delta(t)} = \cos\delta(t)\,\hat{\sigma}_x + \sin\delta(t)\,\hat{\sigma}_y
σ^δ(t)=cos⁡δ(t) σ^x+sin⁡δ(t) σ^y\hat{\sigma}_{\delta(t)} = \cos\delta(t)\,\hat{\sigma}_x + \sin\delta(t)\,\hat{\sigma}_y

Stroboscopic displacement coupling can be time-varying

JD, Siddiqi group, PRL 120, 020505 (2018)

Jumps : Faster drag speeds allow trajectories to jump to the opposite pole, decreasing ensemble-averaged dragging fidelity

Jump-axis : Dragging dynamics causes lag of actual Zeno-pinned behind the measurement axis by a fixed angle

\Omega = 2\pi\,v : \text{dragging frequency}
Ω=2π v:dragging frequency\Omega = 2\pi\,v : \text{dragging frequency}
\Gamma_D = 1/2\tau_m\eta : \text{measurement dephasing rate}
ΓD=1/2τmη:measurement dephasing rate\Gamma_D = 1/2\tau_m\eta : \text{measurement dephasing rate}

JD, Siddiqi group, PRL 120, 020505 (2018)

Pinned to poles : Other than the jumps, state remain pinned to lagged measurement poles

Incoherent Zeno-Dragging Qubit Gate

\Omega = 2\pi\,v : \text{dragging frequency}
Ω=2π v:dragging frequency\Omega = 2\pi\,v : \text{dragging frequency}
\Gamma_D = 1/2\tau_m\eta : \text{measurement dephasing rate}
ΓD=1/2τmη:measurement dephasing rate\Gamma_D = 1/2\tau_m\eta : \text{measurement dephasing rate}

State collapses to jump-axis

JD, Siddiqi group, PRL 120, 020505 (2018)

Incoherent Zeno-Dragging Qubit Gate

Post-selecting on trajectories with an average readout with a value >1 keeps only trajectories that did not jump, heralding a reasonably high-fidelity dragging gate for that subset

 

Alternatively, the jump may be observed, then corrected later

JD, Siddiqi group, PRL 120, 020505 (2018)

Incoherent Zeno-Dragging Qubit Gate

Siddiqi group, Nature 538, 491 (2016)

4 pumps, symmetrically detuned from 2 resonator modes
 

2 simultaneous noncommuting observables

Partial collapses compete with each other, preventing full collapse to a stationary state

If observables are maximally non-commuting, creates persistent phase-diffusion in Bloch sphere

Noncommuting Observable Measurement

\hat{H}^{\text{rf}}_{\text{q-r}} = \chi_1\,(\hat{a}+\hat{a}^\dagger)\,\hat{\sigma}_{\delta_1} + \chi_2\,(\hat{b}+\hat{b}^\dagger)\,\hat{\sigma}_{\delta_2}
H^q-rrf=χ1 (a^+a^†) σ^δ1+χ2 (b^+b^†) σ^δ2\hat{H}^{\text{rf}}_{\text{q-r}} = \chi_1\,(\hat{a}+\hat{a}^\dagger)\,\hat{\sigma}_{\delta_1} + \chi_2\,(\hat{b}+\hat{b}^\dagger)\,\hat{\sigma}_{\delta_2}
\hat{\sigma}_{\delta_k} = \cos\delta_k\,\hat{\sigma}^{\text{rf}}_x + \sin\delta_k\,\hat{\sigma}^{\text{rf}}_y
σ^δk=cos⁡δk σ^xrf+sin⁡δk σ^yrf\hat{\sigma}_{\delta_k} = \cos\delta_k\,\hat{\sigma}^{\text{rf}}_x + \sin\delta_k\,\hat{\sigma}^{\text{rf}}_y

Siddiqi group, Nature 538, 491 (2016)

State purifies, but diffuses randomly

Basins of attraction if measurement axes are nearly aligned

Noncommuting Observable Measurement

Siddiqi group, Nature 538, 491 (2016)

State disturbance can be measured

Result agrees with the lower bound set by the Maccone-Pati relation involving the sum of variances:

Uncertainty relation forces the random state diffusion when measuring incompatible axes

Noncommuting Observable Measurement

Circuit Quantum Electrodynamics

Ideas:

  • Size scale of device (\(\mu\)m) is smaller than the characteristic wavelength of superconducting Cooper pairs: quantum coherence is relevant to the motion
     
  • Cooper pairs (bosons) condense into collective charge motion that is well described by an effective "center of charge", much like rigid bodies are described by an effective center of mass in classical mechanics
     
  • Collective charge motion along superconducting wires can be described by the currents and voltages it produces, which are then easily related to measurable  capacitances, inductances, and impedances
     
  • Quantization of the conjugate variables of flux and charge follows from the usual quantization of the electromagnetic field

How do we model collective mesoscopic quantum coherence?

Circuit QED (Technical Details)

Definitions:

  • Map hardware into a circuit of nodes
    connected by branches (e.g., capacitor, inductor, etc.)
     
  • Define voltage and current for each branch via EM fields,
    as well as the flux and charge stored in each element:




     
  • Define ground node and tree of active nodes connecting both capacitors and inductors - the fluxes to ground are the dynamical circuit variables

Vool, U., and Devoret, M. (2017) . doi: 10.1002/cta.2359.

v_b(t) = \int_b \vec{E}(\vec{r},t)\cdot d\vec{\ell}
vb(t)=∫bE⃗(r⃗,t)⋅dℓ⃗v_b(t) = \int_b \vec{E}(\vec{r},t)\cdot d\vec{\ell}
i_b(t) = \frac{1}{\mu_0}\oint_b \vec{B}(\vec{r},t)\cdot d\vec{s}
ib(t)=1μ0∮bB⃗(r⃗,t)⋅ds⃗i_b(t) = \frac{1}{\mu_0}\oint_b \vec{B}(\vec{r},t)\cdot d\vec{s}
\Phi_b(t) = \int_{-\infty}^t v(t')dt'
Φb(t)=∫−∞tv(t′)dt′\Phi_b(t) = \int_{-\infty}^t v(t')dt'
Q_b(t) = \int_{-\infty}^t i(t')dt'
Qb(t)=∫−∞ti(t′)dt′Q_b(t) = \int_{-\infty}^t i(t')dt'
\phi_n(t) = \sum_{b\in\mathcal{B}}\Phi_b(t)
ϕn(t)=∑b∈BΦb(t)\phi_n(t) = \sum_{b\in\mathcal{B}}\Phi_b(t)

Branches \(b\in\mathcal{B}\) in path connecting node \(n\) to ground through capacitors

\mathcal{L} = \mathcal{E}_{\text{kin}}[\{\dot{\phi}_n\}] - \mathcal{E}_{\text{pot}}\left[\{\phi_n\}\right]
L=Ekin[{ϕ˙n}]−Epot[{ϕn}]\mathcal{L} = \mathcal{E}_{\text{kin}}[\{\dot{\phi}_n\}] - \mathcal{E}_{\text{pot}}\left[\{\phi_n\}\right]
\displaystyle q_n \equiv \frac{\partial \mathcal{L}}{\partial \dot{\phi}_n}
qn≡∂L∂ϕ˙n\displaystyle q_n \equiv \frac{\partial \mathcal{L}}{\partial \dot{\phi}_n}

charge conjugate to node flux \(\phi_n\)

\{ \phi_n,\ q_n \}_{\text{PB}} = \pm 1
{ϕn, qn}PB=±1\{ \phi_n,\ q_n \}_{\text{PB}} = \pm 1

(\(+1\) capacitive, \(-1\) inductive)

  • Primary circuit elements:
    capacitor, inductor, and Josephson junction


     

Vool, U., and Devoret, M. (2017) . doi: 10.1002/cta.2359.

i_L(t) = \frac{\Phi(t) - \Phi_0}{L}
iL(t)=Φ(t)−Φ0Li_L(t) = \frac{\Phi(t) - \Phi_0}{L}
v_C(t) = \frac{Q(t) - Q_0}{C}
vC(t)=Q(t)−Q0Cv_C(t) = \frac{Q(t) - Q_0}{C}
i_{J}(t) = I_0\,\sin\left(\frac{\Phi(t) - \Phi_0}{\hbar/2e}\right)
iJ(t)=I0 sin⁡(Φ(t)−Φ0ℏ/2e)i_{J}(t) = I_0\,\sin\left(\frac{\Phi(t) - \Phi_0}{\hbar/2e}\right)
\mathcal{E}_C = \frac{(Q - Q_0)^2}{2C} = \frac{C\,\dot{\phi}^2}{2} = \frac{q^2}{2C}
EC=(Q−Q0)22C=C ϕ˙22=q22C\mathcal{E}_C = \frac{(Q - Q_0)^2}{2C} = \frac{C\,\dot{\phi}^2}{2} = \frac{q^2}{2C}
\mathcal{E}_L = \frac{(\phi - \phi_0)^2}{2L}
EL=(ϕ−ϕ0)22L\mathcal{E}_L = \frac{(\phi - \phi_0)^2}{2L}
\mathcal{E}_J = \frac{(\hbar/2e)^2}{L_J}\left[1 - \cos\left(\frac{(\phi - \phi_0)}{\hbar/2e}\right)\right]
EJ=(ℏ/2e)2LJ[1−cos⁡((ϕ−ϕ0)ℏ/2e)]\mathcal{E}_J = \frac{(\hbar/2e)^2}{L_J}\left[1 - \cos\left(\frac{(\phi - \phi_0)}{\hbar/2e}\right)\right]
L_J = \frac{(\hbar/2e)}{I_0}
LJ=(ℏ/2e)I0L_J = \frac{(\hbar/2e)}{I_0}
\{
{\{

"Kinetic" energy:

"Potential" energy:

  • Canonical quantization : (equivalent to quantizing \(\vec{E},\vec{B}\) )
\{\phi,\,q\}_{\text{PB}} \to \frac{1}{i\hbar}[\hat{\phi},\,\hat{q}]
{ϕ, q}PB→1iℏ[ϕ^, q^] \{\phi,\,q\}_{\text{PB}} \to \frac{1}{i\hbar}[\hat{\phi},\,\hat{q}]
\hat{a} \equiv \frac{1}{\sqrt{2}}(\frac{\hat{\phi}}{\phi_z} + i \frac{\hat{q}}{q_z})
a^≡12(ϕ^ϕz+iq^qz)\hat{a} \equiv \frac{1}{\sqrt{2}}(\frac{\hat{\phi}}{\phi_z} + i \frac{\hat{q}}{q_z})
\phi_z\,q_z = \hbar
ϕz qz=ℏ\phi_z\,q_z = \hbar
\phi_z / q_z \equiv Z_0
ϕz/qz≡Z0\phi_z / q_z \equiv Z_0
[\hat{a},\,\hat{a}^\dagger] = \hat{1}
[a^, a^†]=1^[\hat{a},\,\hat{a}^\dagger] = \hat{1}

Circuit QED (Technical Details)

(An)harmonic Oscillators

  • Resonator:

Vool, U., and Devoret, M. (2017) . doi: 10.1002/cta.2359.

\hat{H} = \frac{\hat{q}_J^2}{2C_J} + \frac{(\hbar/2e)^2}{L_J}\left[1 - \cos\left(\frac{\hat{\phi}_J}{\hbar/2e}\right)\right]
H^=q^J22CJ+(ℏ/2e)2LJ[1−cos⁡(ϕ^Jℏ/2e)]\hat{H} = \frac{\hat{q}_J^2}{2C_J} + \frac{(\hbar/2e)^2}{L_J}\left[1 - \cos\left(\frac{\hat{\phi}_J}{\hbar/2e}\right)\right]
L_J = \frac{(\hbar/2e)}{I_0}
LJ=(ℏ/2e)I0L_J = \frac{(\hbar/2e)}{I_0}
\hat{H}_r = \frac{\hat{q}^2}{2C} + \frac{\hat{\phi}^2}{2L} = \hbar\omega_r(\hat{a}^\dagger\hat{a} + \frac{1}{2})
H^r=q^22C+ϕ^22L=ℏωr(a^†a^+12)\hat{H}_r = \frac{\hat{q}^2}{2C} + \frac{\hat{\phi}^2}{2L} = \hbar\omega_r(\hat{a}^\dagger\hat{a} + \frac{1}{2})
\hat{a} \equiv \frac{1}{\sqrt{2}}(\frac{\hat{\phi}}{\phi_z} + i \frac{\hat{q}}{q_z})
a^≡12(ϕ^ϕz+iq^qz)\hat{a} \equiv \frac{1}{\sqrt{2}}(\frac{\hat{\phi}}{\phi_z} + i \frac{\hat{q}}{q_z})
\phi_z\,q_z = \hbar
ϕz qz=ℏ\phi_z\,q_z = \hbar
\phi_z / q_z = \sqrt{\frac{L}{C}}
ϕz/qz=LC\phi_z / q_z = \sqrt{\frac{L}{C}}
\omega_r = \sqrt{\frac{1}{LC}}
ωr=1LC\omega_r = \sqrt{\frac{1}{LC}}
  • Transmon:

Josephson junction shunted by large capacitance:
\(E_J/E_C \sim 100, \; E_J = \frac{(\hbar/2e)^2}{L_J}, \; E_C = \frac{e^2}{2C_J} \)

\hat{H}_q \approx \hbar\omega_q(\hat{b}^\dagger\hat{b} + \frac{1}{2}) - \frac{\hbar\delta_q}{12}(\hat{b}+\hat{b}^\dagger)^4
H^q≈ℏωq(b^†b^+12)−ℏδq12(b^+b^†)4\hat{H}_q \approx \hbar\omega_q(\hat{b}^\dagger\hat{b} + \frac{1}{2}) - \frac{\hbar\delta_q}{12}(\hat{b}+\hat{b}^\dagger)^4
C_J
CJC_J
L_J
LJL_J
\omega_q = \frac{\sqrt{8 E_J E_C}}{\hbar}
ωq=8EJECℏ\omega_q = \frac{\sqrt{8 E_J E_C}}{\hbar}
\delta_q = \frac{E_C}{\hbar}
δq=ECℏ\delta_q = \frac{E_C}{\hbar}
\hat{b} \equiv \frac{1}{\sqrt{2}}(\frac{\hat{\phi}_J}{\phi_{J,z}} + i \frac{\hat{q}_J}{q_{J,z}})
b^≡12(ϕ^JϕJ,z+iq^JqJ,z)\hat{b} \equiv \frac{1}{\sqrt{2}}(\frac{\hat{\phi}_J}{\phi_{J,z}} + i \frac{\hat{q}_J}{q_{J,z}})
\phi_{J,z}\,q_{J,z} = \hbar
ϕJ,z qJ,z=ℏ\phi_{J,z}\,q_{J,z} = \hbar
\phi_{J,z} / q_{J,z} = \sqrt{\frac{L_J}{C_J}}
ϕJ,z/qJ,z=LJCJ\phi_{J,z} / q_{J,z} = \sqrt{\frac{L_J}{C_J}}

Qubit - Resonator Coupling

  • Resonator + Transmon:

Vool, U., and Devoret, M. (2017) . doi: 10.1002/cta.2359.

C_J
CJC_J
L_J
LJL_J
C_c
CcC_c
\hat{H} = \hat{H}_q + \hat{H}_r + \hat{H}_c
H^=H^q+H^r+H^c\hat{H} = \hat{H}_q + \hat{H}_r + \hat{H}_c
g = \frac{C_c\sqrt{\omega_q\omega_r}}{2\sqrt{C_r(C_J+C_c)}}
g=Ccωqωr2Cr(CJ+Cc)g = \frac{C_c\sqrt{\omega_q\omega_r}}{2\sqrt{C_r(C_J+C_c)}}
\hat{H}_c \approx \hbar g (\hat{a} + \hat{a}^\dagger)(\hat{b} + \hat{b}^\dagger)
H^c≈ℏg(a^+a^†)(b^+b^†)\hat{H}_c \approx \hbar g (\hat{a} + \hat{a}^\dagger)(\hat{b} + \hat{b}^\dagger)
\chi = \delta_q\frac{g^2}{\Delta^2}
χ=δqg2Δ2\chi = \delta_q\frac{g^2}{\Delta^2}

Dispersive approximation, including rotating-wave approx (RWA):

|\Delta| \equiv |\omega_q - \omega_r| \gg |g|
∣Δ∣≡∣ωq−ωr∣≫∣g∣|\Delta| \equiv |\omega_q - \omega_r| \gg |g|
\hat{H}_c \approx 2\chi\,\hat{b}^\dagger\hat{b}\,\hat{a}^\dagger\hat{a} \approx \chi\,\hat{\sigma}_z\,\hat{a}^\dagger\hat{a}
H^c≈2χ b^†b^ a^†a^≈χ σ^z a^†a^\hat{H}_c \approx 2\chi\,\hat{b}^\dagger\hat{b}\,\hat{a}^\dagger\hat{a} \approx \chi\,\hat{\sigma}_z\,\hat{a}^\dagger\hat{a}
\hat{H}_q \approx \hbar\omega_q \frac{\hat{\sigma}_z}{2}
H^q≈ℏωqσ^z2\hat{H}_q \approx \hbar\omega_q \frac{\hat{\sigma}_z}{2}
\omega_r \mapsto \omega_r \pm \chi
ωr↦ωr±χ\omega_r \mapsto \omega_r \pm \chi

Resonator frequency depends
on transmon energy levels

\Rightarrow
⇒\Rightarrow

Transmission Line Coupling

  • Resonator + Transmon + Transmission Line:

Vool, U. , and Devoret, M.  ( 2017 ) . doi: 10.1002/cta.2359 .

C_J
CJC_J
L_J
LJL_J
C_c
CcC_c
\hat{H} = \hat{H}_q + \hat{H}_r + \hat{H}_c + \hat{H}_t + \hat{H}_{rt}
H^=H^q+H^r+H^c+H^t+H^rt\hat{H} = \hat{H}_q + \hat{H}_r + \hat{H}_c + \hat{H}_t + \hat{H}_{rt}
\displaystyle \hat{H}_t = \int_0^\lambda (\frac{\hat{q}(x)^2}{2c} + \frac{(\partial_x\hat{\phi}(x))^2}{2l})dx = v \int_0^\lambda(\hat{A}^\leftarrow(x)^2 + \hat{A}^\rightarrow(x)^2)dx = \sum_{k=-\infty}^\infty\hbar\omega_k\frac{\hat{c}_k^\dagger \hat{c}_k + \hat{c}_k \hat{c}_k^\dagger}{2}
H^t=∫0λ(q^(x)22c+(∂xϕ^(x))22l)dx=v∫0λ(A^←(x)2+A^→(x)2)dx=∑k=−∞∞ℏωkc^k†c^k+c^kc^k†2\displaystyle \hat{H}_t = \int_0^\lambda (\frac{\hat{q}(x)^2}{2c} + \frac{(\partial_x\hat{\phi}(x))^2}{2l})dx = v \int_0^\lambda(\hat{A}^\leftarrow(x)^2 + \hat{A}^\rightarrow(x)^2)dx = \sum_{k=-\infty}^\infty\hbar\omega_k\frac{\hat{c}_k^\dagger \hat{c}_k + \hat{c}_k \hat{c}_k^\dagger}{2}
C_{rt}
CrtC_{rt}

(OUT: to amplifier and detector)

(IN: from signal generator)

Campagne-Ibarcq, P.,  Ph.D. thesis  ( 2017 ) .

[\hat{\phi}_t(x),\hat{q}_t(x')] = i\hbar\delta(x-x')
[ϕ^t(x),q^t(x′)]=iℏδ(x−x′)[\hat{\phi}_t(x),\hat{q}_t(x')] = i\hbar\delta(x-x')
[\hat{A}^\rightarrow(x),\hat{A}^\rightarrow(x')] = \frac{i\hbar}{2}\partial_x\delta(x-x')
[A^→(x),A^→(x′)]=iℏ2∂xδ(x−x′)[\hat{A}^\rightarrow(x),\hat{A}^\rightarrow(x')] = \frac{i\hbar}{2}\partial_x\delta(x-x')
[\hat{c}_k,\hat{c}^\dagger_{k'}] = \delta_{kk'}
[c^k,c^k′†]=δkk′[\hat{c}_k,\hat{c}^\dagger_{k'}] = \delta_{kk'}
\hat{A}^\rightleftharpoons(x) = \frac{1}{2\sqrt{v}}(\frac{\hat{q}(x)}{\sqrt{c}} \mp \frac{\partial_x\hat{\phi}(x)}{\sqrt{l}})
A^⇌(x)=12v(q^(x)c∓∂xϕ^(x)l)\hat{A}^\rightleftharpoons(x) = \frac{1}{2\sqrt{v}}(\frac{\hat{q}(x)}{\sqrt{c}} \mp \frac{\partial_x\hat{\phi}(x)}{\sqrt{l}})
v = \frac{1}{\sqrt{lc}}
v=1lcv = \frac{1}{\sqrt{lc}}
\displaystyle\hat{c}_k = \sqrt{\frac{2v}{\lambda}}\int_0^\lambda \hat{A}^\rightarrow(x)\frac{e^{-ikx}}{\sqrt{\hbar\omega_k}}dx, \; k>0
c^k=2vλ∫0λA^→(x)e−ikxℏωkdx,  k>0\displaystyle\hat{c}_k = \sqrt{\frac{2v}{\lambda}}\int_0^\lambda \hat{A}^\rightarrow(x)\frac{e^{-ikx}}{\sqrt{\hbar\omega_k}}dx, \; k>0
c,l
c,lc,l

Capacitance, Inductance per unit length

Z_t = \sqrt{\frac{l}{c}}
Zt=lcZ_t = \sqrt{\frac{l}{c}}

Vool, U. , and Devoret, M.  ( 2017 ) . doi: 10.1002/cta.2359 .

C_J
CJC_J
L_J
LJL_J
C_c
CcC_c
\hat{H} = \hat{H}_q + \hat{H}_r + \hat{H}_c + \hat{H}_t + \hat{H}_{rt}
H^=H^q+H^r+H^c+H^t+H^rt\hat{H} = \hat{H}_q + \hat{H}_r + \hat{H}_c + \hat{H}_t + \hat{H}_{rt}
\hat{H}_t = \sum_{k=-\infty}^\infty\hbar\omega_k(\hat{c}_k^\dagger \hat{c}_k + \frac{1}{2})
H^t=∑k=−∞∞ℏωk(c^k†c^k+12)\hat{H}_t = \sum_{k=-\infty}^\infty\hbar\omega_k(\hat{c}_k^\dagger \hat{c}_k + \frac{1}{2})
C_{rt}
CrtC_{rt}

(OUT: to amplifier and detector)

(IN: from signal generator)

Campagne-Ibarcq, P.,  Ph.D. thesis  ( 2017 ) .

\hat{H}_{rt} = \hat{\Phi}\hat{I}(x=0)
H^rt=Φ^I^(x=0)\hat{H}_{rt} = \hat{\Phi}\hat{I}(x=0)
= \frac{\hbar}{2}\sqrt{\frac{Z_r}{Z_t}}(\hat{a}+\hat{a}^\dagger)\sqrt{\frac{v}{\lambda}}\sum_{k=0}^\infty\sqrt{\omega_k}(\hat{c}_{-k} + \hat{c}^\dagger_{-k} -\hat{c}_k - \hat{c}^\dagger_k)
=ℏ2ZrZt(a^+a^†)vλ∑k=0∞ωk(c^−k+c^−k†−c^k−c^k†)= \frac{\hbar}{2}\sqrt{\frac{Z_r}{Z_t}}(\hat{a}+\hat{a}^\dagger)\sqrt{\frac{v}{\lambda}}\sum_{k=0}^\infty\sqrt{\omega_k}(\hat{c}_{-k} + \hat{c}^\dagger_{-k} -\hat{c}_k - \hat{c}^\dagger_k)
v = \frac{1}{\sqrt{lc}}
v=1lcv = \frac{1}{\sqrt{lc}}
Z_t = \sqrt{\frac{l}{c}}
Zt=lcZ_t = \sqrt{\frac{l}{c}}
\equiv \frac{\hbar}{2}i\sqrt{\kappa}(\hat{a}+\hat{a}^\dagger)(\hat{c}_{\text{in}}+\hat{c}_{\text{in}}^\dagger - \hat{c}_{\text{out}} - \hat{c}_{\text{out}}^\dagger)
≡ℏ2iκ(a^+a^†)(c^in+c^in†−c^out−c^out†)\equiv \frac{\hbar}{2}i\sqrt{\kappa}(\hat{a}+\hat{a}^\dagger)(\hat{c}_{\text{in}}+\hat{c}_{\text{in}}^\dagger - \hat{c}_{\text{out}} - \hat{c}_{\text{out}}^\dagger)
\kappa \equiv \omega_r \frac{Z_r}{Z_t}
κ≡ωrZrZt\kappa \equiv \omega_r \frac{Z_r}{Z_t}
\hat{c}_{\text{in}} \equiv -i\sqrt{\frac{v}{\lambda}}\sum_{k=0}^\infty\sqrt{\frac{\omega_k}{\omega_r}}\,\hat{c}_{-k}
c^in≡−ivλ∑k=0∞ωkωr c^−k\hat{c}_{\text{in}} \equiv -i\sqrt{\frac{v}{\lambda}}\sum_{k=0}^\infty\sqrt{\frac{\omega_k}{\omega_r}}\,\hat{c}_{-k}
\hat{c}_{\text{out}} \equiv -i\sqrt{\frac{v}{\lambda}}\sum_{k=0}^\infty\sqrt{\frac{\omega_k}{\omega_r}}\,\hat{c}_{k}
c^out≡−ivλ∑k=0∞ωkωr c^k\hat{c}_{\text{out}} \equiv -i\sqrt{\frac{v}{\lambda}}\sum_{k=0}^\infty\sqrt{\frac{\omega_k}{\omega_r}}\,\hat{c}_{k}

Resonator decay rate near \(\omega_r\)

Boundary condition:

\hat{\Phi}_r = \hat{\phi}_t(x=0)
Φ^r=ϕ^t(x=0)\hat{\Phi}_r = \hat{\phi}_t(x=0)
\Rightarrow \sqrt{\kappa}\,\hat{a} - \hat{c}_{\text{in}} = \hat{c}_{\text{out}}
⇒κ a^−c^in=c^out\Rightarrow \sqrt{\kappa}\,\hat{a} - \hat{c}_{\text{in}} = \hat{c}_{\text{out}}

"Input-output relation"

Transmission Line Coupling