Electromagnetically induced transparency in superconducting quantum circuits: Effects of decoherence, tunneling, and multilevel crosstalk

We explore theoretically electromagnetically induced transparency (EIT) in a superconducting quantum circuit (SQC). The system is a persistent-current flux qubit biased in a $\Lambda$ configuration. Previously [Phys. Rev. Lett. 93, 087003 (2004)], we showed that an ideally prepared EIT system provides a sensitive means to probe decoherence. Here, we extend this work by exploring the effects of imperfect dark-state preparation and specific decoherence mechanisms (population loss via tunneling, pure dephasing, and incoherent population exchange). We find an initial, rapid population loss from the $\Lambda$ system for an imperfectly prepared dark state. This is followed by a slower population loss due to both the detuning of the microwave fields from the EIT resonance and the existing decoherence mechanisms. We find analytic expressions for the slow loss rate, with coefficients that depend on the particular decoherence mechanisms, thereby providing a means to probe, identify, and quantify various sources of decoherence with EIT. We go beyond the rotating wave approximation to consider how strong microwave fields can induce additional off-resonant transitions in the SQC, and we show how these effects can be mitigated by compensation of the resulting ac Stark shifts.

Figure 1

A PC qubit with a dc SQUID measuring device. (a) A PC qubit, a superconducting loop with two Josephson junctions of equal dimension and the third scaled by a factor $\alpha$, as shown in the inner loop. The outer loop is a dc SQUID, which is used to measure the magnetic moment of the qubit. (b) A representation of the potential energy of the PC qubit as a function of $f$, the magnetic flux in the loop in units of the flux quantum ${\Phi }_0$. The qubit potential can range from an asymmetric double well biased to the right to a symmetric double well and to a symmetric double well biased to the left for $f$ ranging from $<1∕2$, $=1∕2$, and $>1∕2$, respectively. (c) One-dimensional double-well potential and energy-level diagram for $f=0.502$, in which case we have a three-level system in the left-hand well. States $\mid 1⟩$ and $\mid 2⟩$ are metastable, while $\mid 3⟩$ will have significant loss via resonant tunneling to $\mid 4⟩$ $\left({\sigma }_{34}\right)$. The right-hand well states undergo fast damping $({\Gamma }_4,{\Gamma }_5,{\Gamma }_6)$ via the SQUID measurement and intrawell relaxation to lower states. Coupling between our three levels is induced by two resonant microwave fields with Rabi frequencies ${\Omega }_{13}$ and ${\Omega }_{23}$, forming the $\Lambda$ system. The qubit parameters we use in calculations are ${\omega }_2-{\omega }_1=\left(2\pi \right)27.8\mathrm{GHz}$ and ${\omega }_3-{\omega }_2=\left(2\pi \right)27\mathrm{GHz}$, with matrix elements $x_{ij}$ for $(i,j)=(1,2),(2,3)$, and $(1,3)$ set to $-0.0145$, $-0.0371$, and $-0.0263$, respectively.

Figure 2

(Color online) Suppression of tunneling due to EIT for various ideal wavefunctions. (a) The populations of the states as a function of time in the presence of applied fields ${\Omega }_{13}={\Omega }_{23}=\left(2\pi \right)150\mathrm{MHz}$ and tunneling rate ${\Gamma }_3^{\left(t\right)}=\left(2\pi \right)130\mathrm{MHz}=1∕1.2\mathrm{ns}$ for the initial state ${\rho }_{11}={\rho }_{22}=0.5,{\rho }_{12}=-0.5$ (the dark state). The dotted (red) curve shows ${\rho }_{11}$, the dashed (blue) curve ${\rho }_{22}$, and the thin solid (green) curve ${\rho }_{33}$. The total population (sum of the three) is the thick solid (black) curve. There is a slow exponential decay of the population due to the dephasing rate ${\gamma }_{12}=\left(2\pi \right)1\mathrm{MHz}$. (b) The population evolutions (same convention) for the initial state ${\rho }_{11}={\rho }_{22}=0.5,{\rho }_{12}=0.5$ (the absorbing state). (c) The population evolutions for an initial state $\mid 1⟩$ (which is an equal superposition of the dark and absorbing states).

Figure 3

(Color online) Imperfect state preparation. (a) The time evolution of the population in the left well as a function of initial state of the form $\mid {\Psi }_{\mathrm{init}}⟩=\sqrt{p_1}\mid 1⟩-\sqrt{(1-p_1)}\mid 2⟩$. The uppermost curve is for the dark state $p_1=0.5$. Successively lower curves are for $p_1=0.6,0.7,0.8,0.9$, and $1.0$. There is a sharp initial decay when there are deviations from the dark state. Inset: the population in the left well at $200\mathrm{ns}$ versus $p_1$. (b) The population decay out of the left well as a function of the initial phase of the prepared state, $\mid {\Psi }_{\mathrm{init}}⟩=(\mid 1⟩-e^{i\theta }\mid 2⟩)∕\sqrt{2}$. $\theta =0$ (top curve) and the other curves are $\pi ∕5$, $2\pi ∕5$, $3\pi ∕5$, $4\pi ∕5$, and $\pi$. We see full decay for $\theta =\pi$, the absorbing state $\mid {\Psi }_A⟩=(\mid 1⟩+\mid 2⟩)∕\sqrt{2}$. Inset: the population in the left well at $200\mathrm{ns}$ as a function of $\theta$.

Figure 4

(Color online) EIT in the presence of detuning. (a) Numerical calculation of the total population in time when ${\Delta }_{13}=0\mathrm{MHz}$ and at various ${\Delta }_{23}$ (top to bottom curve) $\left(2\pi \right)$ 0, 10, 20, 30, $40\mathrm{MHz}$. For ${\Delta }_{23}=0$, the decay is due to pure dephasing, while the decay is sharper when ${\Delta }_{23}\ne 0$. Inset: the population in the left well at $200\mathrm{ns}$ as a function of the two-photon detuning ${\Delta }_2\equiv {\Delta }_{13}-{\Delta }_{23}$, with ${\Delta }_{13}=0$ (circles). The solid curve shows the prediction (13). The diamonds show the case ${\Delta }_{13}=\left(2\pi \right)$ $1\mathrm{GHz}$, varying ${\Delta }_2$ about the two-photon resonance. (b) The population decay as a function one-photon detuning at two-photon resonance, ${\Delta }_{13}={\Delta }_{23}=\Delta$ for (top to bottom) $\left(2\pi \right)$ 3, 2, 1, 0.5, $0\mathrm{GHz}$. As the one-photon detuning increases, the effective coupling of the fields to the transition reduces, hence reducing the rate of decay. Inset: the population in the left well at $200\mathrm{ns}$ as a function of the detuning for the dark state (circles). For comparison we also show the population remaining for the absorbing state $\mid {\Psi }_{\mathrm{init}}⟩=(\mid 1⟩+\mid 2⟩)∕\sqrt{2}$ at the same detunings (diamonds).

Figure 5

(Color online) Consequences of the measurement state characteristics. (a) The thinner curves (blue) show populations ${\rho }_{44}$ (solid curves) and ${\rho }_{33}$ (dashed curves) as a function of time for a fast readout ${\Gamma }_4^{-1}=1\mathrm{ns}<{\sigma }_{34}^{-1}$ (with scale on the right side). After initial transient period, the two values reach quasi-steady-state values, which undergo slow exponential decay. Conversely the thick (red online) curves show a slow measurement case ${\Gamma }_4^{-1}=25\mathrm{ns}⪢{\sigma }_{34}^{-1}$ (scale on the left). In this case the populations do not reach a quasi steady state over the time scale plotted. (b) The total population remaining versus time for the same two cases. (c) The population remaining at $100\mathrm{ns}$ for varying measurement rates (dots) and compared to the prediction of a three-level system with ${\Gamma }_3^{\left(t\right)}$. As the measurement gets slower, this prediction slightly and increasingly underestimates the actual population which should be observed. (d) Numerical results (dots) and three-level model predictions (solid curves) now letting ${\delta }_{34}$ vary, for the cases ${\Gamma }_4^{-1}=1\mathrm{ns}$ (black, lower curve) and $10\mathrm{ns}$ (upper, red curve).

Figure 6

(Color online) EIT loss due to pure dephasing. (a) The population ${\rho }_{33}$ versus time in the presence of a pure dephasing ${\gamma }_{12}=\left(2\pi \right)$ $1\mathrm{MHz}$ (solid, red curve, scale on left) and ${\gamma }_{12}=\left(2\pi \right)$ $20\mathrm{MHz}$ (dashed, blue curve, scale on right). In the slow dephasing case ${\rho }_{33}$ quickly reaches a plateau value ${\rho }_{33}^{\left(\max \right)}$ and then undergoes a slow exponential decay. For the faster dephasing, the exponential decay time is similar to the time required to reach the maximum value. In each case, the initial state is ${\rho }_{11}={\rho }_{22}=0.5$ and ${\rho }_{12}=-0.5$, ${\Omega }_{13}={\Omega }_{23}=\left(2\pi \right)$ $150\mathrm{MHz}$, and ${\Gamma }_3^{\left(t\right)}=\left(2\pi \right)$ $130\mathrm{MHz}$. (b) The population decay with varying dephasing rates (top curve to bottom curve) ${\gamma }_{12}=\left(2\pi \right)$ 1, 2, 3, 4, and $5\mathrm{MHz}$. Inset: the population in the left well at $200\mathrm{ns}$ versus ${\gamma }_{12}$ (dots). The solid curve shows the analytic prediction $\exp (-R_L^{\left({\gamma }_{12}\right)}t)$ based on Eq. (17), demonstrating how the population loss can be used as a probe of ${\gamma }_{12}$. (c) Population remaining at $100\mathrm{ns}$ versus detuning ${\Delta }_{13}$ (keeping ${\Delta }_{23}=0)$ for two different field intensities. In these simulations we used the initial conditions ${\rho }_{11}=0.61$, ${\rho }_{22}=0.39$, ${\rho }_{12}=-\sqrt{{\rho }_{11}{\rho }_{22}}$, the fields ${\Omega }_{13}=0.8{\Omega }_{23}$, ${\Gamma }_3^{\left(t\right)}=\left(2\pi \right)$ $159\mathrm{MHz}$, ${\gamma }_{12}=\left(2\pi \right)$ $1\mathrm{MHz}$. The solid curves are the analytic prediction described in the text for ${\Omega }_{23}=\left(2\pi \right)$ $100\mathrm{MHz}$ (blue, upper curve), and $30\mathrm{MHz}$ (black, lower curve). The dots show the numerical results.

Figure 7

(Color online) EIT loss with dephasing, open loss, and closed loss. The population loss after $100\mathrm{ns}$ for several types of decoherence. The parameters are as in Fig. 6c but with ${\Omega }_{23}=\left(2\pi \right)$ $150\mathrm{MHz}$. The curves show the analytic predictions (17, 18), and the dots show numerical results. The lowest curve is for purely open loss ${\Gamma }_2^{\left(t\right)}$, the middle curve for pure dephasing ${\gamma }_{12}$, and the top curve shows pure population exchange ${\Gamma }_{2\to 1}$. The horizontal axis shows the corresponding decoherence rate for each case: ${\Gamma }_2^{\left(t\right)}∕2,{\gamma }_{12},{\Gamma }_{2\to 1}∕2$, respectively.

Figure 8

(Color online) Loss due to resonant tunneling to the right well. In the simulations we assume a (resonant) tunneling rate ${\sigma }_{25}=\left(2\pi \right)$ $5\mathrm{MHz}$, with fields ${\Omega }_{13}=0.8{\Omega }_{23}$ and the corresponding dark state ${\rho }_{11}=0.61$, ${\rho }_{22}=0.39$, ${\rho }_{12}=-\sqrt{{\rho }_{11}{\rho }_{22}}$. (a) The population remaining at $100\mathrm{ns}$ of fields applied with strength ${\Omega }_{23}=50\mathrm{MHz}$, versus the detuning ${\Delta }_{13}$ (keeping ${\Delta }_{23}=0$), but varying the relaxation time ${\Gamma }_5^{-1}$. For the small solid dots (black) ${\Gamma }_5^{-1}=2\mathrm{ns}$. The solid curve shows the analytic prediction based on the effective loss rate ${\Gamma }_2^{\left(t\right)}$ described in the text. The open (blue) dots show a case ${\Gamma }_5^{-1}=15\mathrm{ns}$, in which case a splitting appears in the resonance, contrary to the analytic prediction (dashed curve). The large (red) dots show the case ${\Gamma }_5^{-1}=80\mathrm{ns}$, for which the splitting becomes more pronounced and the loss rate quite small, while the ${\Gamma }_2^{\left(t\right)}$ model predicts complete loss of the population. (b) The population remaining at $100\mathrm{ns}$ at the two photon resonance $({\Delta }_{13}={\Delta }_{23}=0)$ versus ${\Gamma }_5^{-1}$. The solid (black) dots show the case ${\Omega }_{13}=\left(2\pi \right)150\mathrm{MHz}$ and the open (red) dots show ${\Omega }_{13}=\left(2\pi \right)50\mathrm{MHz}$. They roughly agree with each other and the ${\Gamma }_2^{\left(t\right)}$ model (solid curve) for ${\Gamma }_5^{-1}⪡{\sigma }_{25}^{-1}=32\mathrm{ns}$. However, for larger ${\Gamma }_5^{-1}$ the loss becomes slower. The inset shows the population ${\rho }_{55}$ for the fast ($5\mathrm{ns}$, solid curve) and slow ($80\mathrm{ns}$, dashed) relaxation times ${\Gamma }_5^{-1}$ (note the different scales). The fast case looks analogous to decoherence [see Fig. 6a], while oscillations occur in the slow case. (c) The population remaining versus the level detuning ${\delta }_{25}$ in the case ${\Gamma }_5^{-1}=8\mathrm{ns}$ (solid, black dots) and $80\mathrm{ns}$ (open, red dots). The solid and dashed curves show the ${\Gamma }_2^{\left(t\right)}$ model predictions.

Figure 9

(Color online) Crosstalk in a ladder system. (a) Schematic of the dominant crosstalk term: field $b$ (resonant with $\mid 2⟩\leftrightarrow \mid 3⟩$) also couples $\mid 1⟩\leftrightarrow \mid 2⟩$, detuned by $0.7\mathrm{GHz}$. (b) This induces fast oscillations of the ground-state populations ${\rho }_{11}$ and ${\rho }_{22}$ (and a slow overall drift) as shown here for the initial state ${\rho }_{11}=0.61$, ${\rho }_{22}=0.39$ with full coherence and ${\Omega }_{13}=0.8{\Omega }_{23}=\left(2\pi \right)$ $120\mathrm{MHz}$ and ${\Gamma }_3^{\left(t\right)}=\left(2\pi \right)$ $159\mathrm{MHz}$. (c) Population remaining at $50\mathrm{ns}$ for ${\Omega }_{23}=\left(2\pi \right)$ $50\mathrm{MHz}$ (small, blue dots), $100\mathrm{MHz}$ (open, green dots), and $150\mathrm{MHz}$ (large, red dots), with the curves showing the analytic predictions based on the ac Stark shifts described in the text.

Figure 10

(Color online) Coupling to an additional excited level. (a) Schematic of off-resonant microwave coupling of each of states $\mid 1⟩$ and $\mid 2⟩$ to an additional level $\mid e⟩$ above the barrier. (b) The population remaining at $100\mathrm{ns}$ for the same initial state and relative field strengths as in Fig. 9. We show the cases ${\Omega }_{23}=\left(2\pi \right)$ $60\mathrm{MHz}$ (small solid, blue), $100\mathrm{MHz}$ (open, green), and $150\mathrm{MHz}$ (large solid, red). The solid curves show the loss and ac Stark shift predicted in the text, Eqs. (25). To isolate and clearly show the effect we have set $\beta =0$ [from Eq. (20)] and used ${\delta }_{3e}=\left(2\pi \right)$ $1.5\mathrm{GHz}$, instead of the $\left(2\pi \right)$ $10\mathrm{GHz}$ we predict for our proposed parameters. We use $x_{14}=0.0054$ and $x_{24}=-0.0437$ and ${\Gamma }_e={\Gamma }_3^{\left(t\right)}=\left(2\pi \right)$ $159\mathrm{MHz}$.