QuantumTrajectoriesofaSuperconductingQubit.md

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Quantum Trajectories of a Superconducting Qubit

Jan 31, 2017

$\require{AMSsymbols}$

The experiments discussed in this thesis use superconducting circuits to study the process of measurement of a quantum system.

These are the first experiments to successfully track diffusive quantum trajectories in a solid state system. Furthermore, these are the first experiments on any system to use quantum state tomography at discrete times along the trajectory to verify that we have faithfully tracked the qubit state.

[toc]

Parameters, acronyms and concepts

• measurement quantum efficiency $\eta$
• HEMT: high electron mobility transistors, usually at 4K, can only achieve $ \eta\sim 1$
• QND: quantum non-demolition measurement: cases no backaction on the measured observable beyond the usual backaction associated with the acquisition of information
• Heisenberg backaction: the act of acquiring information about one observable will necessarily perturb its canonically conjugate observable
• standard quantum limit
• POVM: positive operator-valued measures
• SME: stochastic master equations
• SSE: stochastic schrodinger equations

Continuous quantum measurement

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Example of non-QND measurement: measure $x$ of a HO: $\psi (x)\propto \mathrm{exp}[-m \omega x^2/2\hbar] $. $\Delta p$ increase after the measurement and the time evolution of $x$ is perturbed.

Indirect measurement:

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An indirect measurement will be QND, provided that the probe is not affected by more than one of any set of non-commuting observables of the measured system.

Formal criteria for QND measurement

QND iff $$[q,U]=0,$$ where $q$ is the general coordinate to measure and $U$ is the unitary operator that generates joint evolution of the measured system and the probe over the full measurement time.

In practice, $U$ is hard to calc, hence use a more restrictive sufficient condition: $$ [q,H]=0 $$ assuming no explicit time dependent of the operator $q$. By definition, $[q,H_{\mathrm{probe}}]=0$. If $q$ is conserved during the evolution of the measured system, $ [q, H_{\mathrm{sys}} ] =0$, thus the requirement is $$ [q, H_{\mathrm{int}} ]=0 $$

Partial and projective measurement

SNR, or strength, of a measurement: $$ S \equiv (\frac{\Delta V}{\sigma})^2 = \frac{4T}{\tau} $$

A partial measurement occurs when $ T \lesssim \tau $, a projective measurement: $ T \gg \tau $. Measurement rate: $ \Gamma_{\mathrm{meas}} =1/\tau $

$ \def\bra#1{\left< #1 \right|} \def\ket#1{\left| #1 \right>} \def\braket#1{ \left< #1 \right> } $

Single shot measurement fidelity $F = 1 - P(1|q_{\ket{0}}) - P(0|q_{\ket{1}}) $

Separation fidelity $ F_s $ sets an upper bound on $F$. Reason reducing $F$:

• qubit energy relaxation
• measurement is not entirely QND hence inducing transition

The quantum efficiency of a measurement describes how close it comes to ideal Heisenberg-limited backaction: $$ \eta_m \equiv \frac{\Gamma_m}{\Gamma_{m,\mathrm{ideal}}} $$

For qubit system, $\Gamma_m\propto\S/S_{ \mathrm{ideal}}$, which is related to $ \Delta V $ and $ \sigma $: $$ \eta_m = \left (\frac{\Delta V}{\Delta V_{\mathrm{ideal}}} \right )^2 \left (\frac{\sigma}{\sigma_{\mathrm{ideal}}} \right )^2 = \eta_{col}\eta_{amp} $$ Additional dephasing from environment (qubit's) is described by $\eta_{env} $, hence total efficiency $\eta = \eta_m \eta_{env} $

POVM: a more general set of measurement operators $ {\Omega_m} $ satisfying $ \sum_m \Omega_m^\dagger \Omega_m = I $. The $m$-th result occurs with $P(m) = \mathrm{Tr}[\Omega_m \rho \Omega_m^\dagger] $ and the state $$ \rho_f = \frac{\Omega_m \rho \Omega_m^\dagger}{\mathrm{Tr}[\Omega_m \rho \Omega_m^\dagger]} $$ Index $m$ can be generalized to be continuous and in the qubit case, $$ \Omega_V = \frac{1}{\mathcal N}[ e^{-2k\eta_m \Delta t(1-V)^2} \ket 0 \bra 0 + e^{-2k\eta_m \Delta t(-1-V)^2} \ket 1 \bra 1 ] $$ where $ \mathcal N $ is for normalization and $ k \eta_m \equiv 1/4\tau $ parametrizes the measurement strength (with measurement operator $ \sqrt{k} \sigma_z $). $V$ is rescaled.

Hence the probability for each measurement result: $$ P( \Omega_V ) = \mathrm{Tr} (\Omega_V \rho \Omega_V^\dagger ) = P(\ket{0}) e^{ -4 k \eta_m \Delta t(1-V)^2 } +P(\ket{1}) e^{ -4 k \eta_m \Delta t(-1-V)^2 } $$

Quantum trajectories

If we understand the backaction of an individual measurement, then this understanding can be used to update our knowledge of the quantum state after (and during) measurement. This is essential to applications in measurement-based quantum feedback and control.

Consider a continuous QND measurement of a qubit, the signal $ V(t) $ is broken up to discrete time steps of width $ \Delta t $ and recorded as $ [ V_0, V_1,...,V_{n-1} ] $, where $$ V_i = \frac{1}{\Delta t} \int^{t_i +\Delta t}_{t_i}V(t)dt $$

• quantum jumps are most readily observed when $ \tau \lesssim \Delta t< \tau_{\mathrm{jump}} < T $
• diffusive trajectories are most readily observed when $ \Delta t \ll \tau < T < \tau_{\mathrm{jump}} $

Stochastic master equations

For weak measurement $ \Delta t \ll \tau $, the distribution of single measurement reduce to $$ P(\Omega_{V_i})\approx e^{ -4k \eta_m \Delta t (V_i - \braket{\sigma_z} )^2 } $$ which shows that $V_i$ can be thought as simply a noisy estimate of $\braket{\sigma_z}$: $ V_i = \braket{\sigma_z} + \frac{\Delta W}{\sqrt{8k\eta_m}\Delta t} $

Similarly, $ \Omega_{V_i} $ can also be re-expressed as $ \Omega_{V_i} \propto e^{-2k\eta_m \Delta t (V_i - \sigma_z)^2} $, in addition with $ \ket{ \psi (t+\Delta t) }\propto \Omega_{V_i} \ket{\psi(t)} $, keeping to first order in $ \Delta t $ and after normalization, we get SSE: $$ d \ket{\psi} = \lbrace -k(\sigma_z - \braket{ \sigma_z })^2 dt + \sqrt{2k} ( \sigma_z - \braket{ \sigma_z })dW \rbrace \ket{\psi(t)} $$ and SME: $$ d\rho = -k [ \sigma_z, [\sigma_z , \rho ] ] dt + \sqrt{2k \eta} ( \sigma_z \rho + \rho \sigma_z - 2 \braket{\sigma_z} \rho )dW $$

Bayesian state update:

$$ \frac{\rho_{11}(t)}{\rho_{00}(t)} = \frac{P(1|V_m)}{P(0|V_m)} = \frac{P(1)}{P(0)} \frac{P(V_m|1)}{P(V_m|0)} = \frac{ \rho_{11}(0) }{\rho_{00}(0)} \frac{ \exp [ -( V_m(t) + \Delta V/2 )^2/2 \sigma^2 ] }{\exp [ -( V_m(t) -\Delta V/2)^2 /2 \sigma^2 ]} $$

Superconducting qubits and circuit QED

Almost things you already know:

• Derivation of the transmon qubit Hamiltonian
• J-C Hamiltonian
• Dispersive measurement
• SNR: $$ S \equiv \left ( \frac{\Delta V_{\mathrm{opt}}}{\sigma} \right )^2 = \frac{64 \chi^2 \bar n \eta_m T}{\kappa} $$ where
• standard deviation of the measurement histograms $\sigma = \sqrt{G/\kappa \eta T} $
• separation between the histograms in the limit of small $\chi/\kappa$ is $ V_{\mathrm{opt}} = \sqrt G \Delta X_2 = 8 \chi \sqrt{G \bar n} /\kappa $

Parametric amplifiers and squeezing

Two types of amplification:

• phase-preserving: $ (\Delta X_{1,2}) _ \mathrm{out} = \sqrt G \sqrt{\Delta X_{1,2}+1} $, add extra vacuum noise because of measuring both quadratures
• phase-sensitive: minimum uncertainty input $\rightarrow$ minimum uncertainty output \begin{eqnarray} (\Delta X_1)_ \mathrm{out} &=& \frac{1}{2\sqrt G} \Delta X_1\cr (\Delta X_2)_ \mathrm{out} &=& 2 \sqrt G \Delta X_2 \end{eqnarray}

Josephson parametric amplifiers:

• Hamiltonian: $ H_{\mathrm{JPA}} = \hbar \omega_0 a^\dagger a + \hbar \frac{K}{2} (a^\dagger)^2 a^2 $
• Adding strong drive: $ H_d = \epsilon_d e^{-i \omega_d t} a^\dagger + h.c. $
• Displace $ a \rightarrow \alpha + d $ and ignore higher order terms: $ H_{\mathrm{sys}}^{(2)} = \tilde \Delta d^\dagger d + \frac{\lambda}{2} (d^{\dagger 2}+d^2) $. (superscript means to the second order in $d$) i.e., the squeezing Hamiltonian

Performance: gain $G$, band width $B$ and dynamic range

• $ B \sqrt G \propto \frac{1}{Q} $, hence decrease $Q$
• However, decreasing $Q$ requires higher power to obtain large amplitude, which will excite higher-order non-linearities
• Hence approximately $ Qp \gtrsim 5 $, where $p = L_J/L_{\mathrm{tot}}$ is the participation ratio of the Josephson inductance to the total inductance
• signal power at which the gain is reduced by 1dB is known as the '1dB compression point' and determines the dynamic range of the amp

Experimental setup

• Qubit: 3D transmon, $ T_1 = 30 \mu s \sim 100 \mu s $
• bias up the paramp: single pump, double pump, flux-pumping; balance the two sidebands isn't trivial
• displace the signal before paramp (eliminating the 'dumb signal')

Calibration experiments

• VNA, cavity response: low probe power: $ \omega_{\ket{0}} $; high powers ($\bar n \gg n_{crit} $): bare cavity freq $ \omega_r $
• two-tone spectroscopy:
  • scan pump freq across $\omega_q$
  • at higher pump power, sharp peak appears at $ \omega_{0\rightarrow 2}/2 $, can be used to calc aharmonicity
• time domain measurement:
  • set optimal measurement freq $ \Omega_m = \omega_{\ket{0}} + \chi $
  • set the amplification axis
  • Rabi, obtain $\pi$ pulse length etc. and time scale depending on $ T_1, T_2^* $
  • Ramsey, characterize qubit freq and dephasing time $ T_2^*$, $\pi/2$ pulse required. Oscillation freq is the detuning.
  • $T_1$ measurement, $\pi$ pulse required
• pulse calibration: $\pi/2$ pulse about $x, -y$ axis required. Fix pulse duration and scan gatting pulse voltage setpoint
• heralded state preparation: projective measurement -> ground state? -> use the prepared state
• AC stark shift calibration: $ \Delta \omega = 2 \chi \bar n $, measurement-induced dephasing: $ \Gamma_{\mathrm{MID}} = \frac{ 8 \chi^2 \bar n }{\kappa} $, which both can be measured by a Ramsey measurement

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Tracking individual quantum trajectories

Quadrature-dependent measurement backaction

• for small $\chi/\kappa$, $X_1$ determines field amplitude; $X_2$ determines phase
• amplify $X_2$: squeeze $X_1$, rate of information extraction $ \Gamma_{meas} \equiv 1/\tau = 2 \Gamma_{Heis} = 2 \Gamma_{MID} $
• amplify $X_1$: squeeze away $\hat \sigma_z$ information contained in $X_2$, no measurement backaction

Correlation between measurement outcomes and the qubit state (using Bayesian state update)

• state update equations for two kinds of measurement
• conditional quantum state tomography:
  • Aim: using tomography to verify the prediction on state after measurement
  • Method: do tomography on states after the same measurement with approximately same measurement outcome
• calculate the quantum trajectory:
  • calculate the updated state after the measurement
  • do conditional quantum state tomography on every possible measurement outcome value
  • they agree quite well (Fig. 6.2)
• Trajectories under driven evolution
  • use sequential two-step procedure to update the qubit state after each time step
  • same conditional quantum state tomography
  • obtained good agreement (Fig. 6.7)
• Comparing Bayesian trajectories to SME trajectories
  • agree well (Fig 6.8)

Ensembles of trajectories

This chapter investigate the interplay between measurement backaction and unitary dynamics

Approach:

• solve SME numerically at a large number of times and perform statistical analysis
• action principle

Process to obtain the action (to use action principle):

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	"deterministic qubit state evolution"->"write probability as delta function"->"fourier transform"->"add probability of measurement result"
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Not understand well, anyway they obtained the action $S$ and the resulting ODE: \begin{eqnarray} \dot x &=& - \gamma x + \Omega z - x z r/\tau \cr \dot z &=& \Omega x +(1-z^2)r/\tau \cr \dot p_x &=& \gamma p_x +\Omega p_z + p_x z r / \tau \cr \dot p_z &=& - \Omega p_x +(p_x x+2p_z z-1)r/\tau \end{eqnarray} which has analytic solution (see Eqn 7.13~7.15) for $ \Omega \ne 0 $. So the optimal quantum path is obtained.

From it, the most likely time can be calculated (time when the trajectory evolve to yield maximum $P(z_F|z_I) $).

Distribution of quantum trajectories

• use pre and post selection to create ensemble of system with chosen initial/final state
• Define closeness of any two trajectories
• calculate it between all possible pairs of trajectories and search for $N$ trajectories with lowest average distance

Results:

• distribution of un-driven trajectories obtained, showing most likely path agree with theory prediction. See Fig 7.2
• $P(z_F|z_I=0) $ vs post-selection time agree with "most likely time"
• distribution of driven trajectories obtained, in an intermediate parameter regime between quantum jumps and diffusive trajectories

Quantum efficiency and squeezing

limiting factor of quantum efficiency in the author's case: imperfect squeezing

Possible limitating factor on $\eta_{amp}$:

1. gain too low to overcome the added noise of the HEMT
2. insufficient paramp bandwidth
3. loss in the paramp, dominated by the dielectric losses in the SiN_x capacitors, $Q\sim 5000$
4. paramp could be hotter than the 20mK plate

Questions

Possibly typo?

• P25 footnote
• P29 eqn 3.1, 3.6, 3.7 (missing $i$?)

Questions

• P34, what (or how) is an external phase bias?
• P50, Eqn(4.15), term $ \Delta_d \alpha (d+d^\dagger) $ absorbed into freq?
• P59, how to set relative sideband amplitudes by relative phase of the modulation times on I and Q ports?
• P61, Fig 5.6, where's the reflected signal from the cavity?