Continuous dynamical decoupling and decoherence-free subspaces for qubits with tunable interaction


Protecting quantum states from the decohering effects of the environment is of great importance for the development of quantum computation devices and quantum simulators. Here, we introduce a continuous dynamical decoupling protocol that enables us to protect the entangling gate operation between two qubits from the environmental noise. We present a simple model that involves two qubits which interact with each other with a strength that depends on their mutual distance and generates the entanglement among them, as well as in contact with an environment. The nature of the environment, that is, whether it acts as an individual or common bath to the qubits, is also controlled by the effective distance of qubits. Our results indicate that the introduced continuous dynamical decoupling scheme works well in protecting the entangling operation. Furthermore, under certain circumstances, the dynamics of the qubits naturally led them into a decoherence-free subspace which can be used complimentary to the continuous dynamical decoupling.

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  1. 1.

    Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715 (2003).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Schlosshauer, M.: Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys. 76, 1267 (2005).

    ADS  Article  Google Scholar 

  3. 3.

    Montanaro, A.: Quantum algorithms: an overview. npj Quantum Inf. 2(15023), 15023 (2015).

    ADS  Article  Google Scholar 

  4. 4.

    Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793 (1996).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Shor, P.W.: Fault-tolerant quantum computation. In: Proceedings of the 37th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, Washington, DC, USA, 1996), FOCS ’96, p. 56

  6. 6.

    Lidar, D.A., Chuang, I.L., Whaley, K.B.: Decoherence-free subspaces for quantum computation. Phys. Rev. Lett. 81, 2594 (1998).

    ADS  Article  Google Scholar 

  7. 7.

    Lidar, D.A., Whaley, K.B.: Decoherence-free subspaces and subsystems. In: Benatti, F., Floreanini, R. (eds.) Irreversible Quantum Dynamics. Lecture Notes in Physics, vol. 622, pp. 83–120. Springer, Berlin (2003)

    Chapter  Google Scholar 

  8. 8.

    Viola, L., Lloyd, S.: Dynamical suppression of decoherence in two-state quantum systems. Phys. Rev. A 58, 2733 (1998).

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Viola, L., Knill, E., Lloyd, S.: Dynamical decoupling of open quantum systems. Phys. Rev. Lett. 82, 2417 (1999).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Zanardi, P.: Symmetrizing evolutions. Phys. Lett. A 258(2), 77 (1999).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Byrd, M.S., Lidar, D.A.: Bang-bang operations from a geometric perspective. Quantum Inf. Process. 1(1), 19 (2002).

    MathSciNet  Article  Google Scholar 

  12. 12.

    Facchi, P., Tasaki, S., Pascazio, S., Nakazato, H., Tokuse, A., Lidar, D.A.: Control of decoherence: analysis and comparison of three different strategies. Phys. Rev. A 71, 022302 (2005).

    ADS  Article  Google Scholar 

  13. 13.

    Khodjasteh, K., Lidar, D.A.: Rigorous bounds on the performance of a hybrid dynamical-decoupling quantum-computing scheme. Phys. Rev. A 78, 012355 (2008).

    ADS  Article  Google Scholar 

  14. 14.

    Khodjasteh, K., Viola, L.: Dynamically error-corrected gates for universal quantum computation. Phys. Rev. Lett. 102, 080501 (2009).

    ADS  Article  Google Scholar 

  15. 15.

    Khodjasteh, K., Viola, L.: Dynamical quantum error correction of unitary operations with bounded controls. Phys. Rev. A 80, 032314 (2009).

    ADS  Article  Google Scholar 

  16. 16.

    Khodjasteh, K., Lidar, D.A., Viola, L.: Arbitrarily accurate dynamical control in open quantum systems. Phys. Rev. Lett. 104, 090501 (2010).

    ADS  Article  Google Scholar 

  17. 17.

    Lidar, D.A.: Review of decoherence free subspaces, noiseless subsystems, and dynamical decoupling. Adv. Chem. Phys. 154, 295 (2012). arXiv:1208.5791

  18. 18.

    Biercuk, M.J., Uys, H., VanDevender, A.P., Shiga, N., Itano, W.M., Bollinger, J.J.: Optimized dynamical decoupling in a model quantum memory. Nature 458(7241), 996 (2009).

    ADS  Article  Google Scholar 

  19. 19.

    Du, J., Rong, X., Zhao, N., Wang, Y., Yang, J., Liu, R.: Preserving electron spin coherence in solids by optimal dynamical decoupling. Nature 461(7268), 1265 (2009).

    ADS  Article  Google Scholar 

  20. 20.

    Damodarakurup, S., Lucamarini, M., Di Giuseppe, G., Vitali, D., Tombesi, P.: Experimental inhibition of decoherence on flying qubits via “bang-bang” control. Phys. Rev. Lett. 103, 040502 (2009).

    ADS  Article  Google Scholar 

  21. 21.

    De Lange, G., Wang, Z., Riste, D., Dobrovitski, V., Hanson, R.: Universal dynamical decoupling of a single solid-state spin from a spin bath. Science 330(6000), 60 (2010).

    ADS  Article  Google Scholar 

  22. 22.

    Souza, A.M., Álvarez, G.A., Suter, D.: Robust dynamical decoupling for quantum computing and quantum memory. Phys. Rev. Lett. 106, 240501 (2011).

    ADS  Article  Google Scholar 

  23. 23.

    Naydenov, B., Dolde, F., Hall, L.T., Shin, C., Fedder, H., Hollenberg, L.C.L., Jelezko, F., Wrachtrup, J.: Dynamical decoupling of a single-electron spin at room temperature. Phys. Rev. B 83, 081201 (2011).

    ADS  Article  Google Scholar 

  24. 24.

    Van der Sar, T., Wang, Z., Blok, M., Bernien, H., Taminiau, T., Toyli, D., Lidar, D., Awschalom, D., Hanson, R., Dobrovitski, V.: Decoherence-protected quantum gates for a hybrid solid-state spin register. Nature 484(7392), 82 (2012).

    ADS  Article  Google Scholar 

  25. 25.

    Fonseca-Romero, K.M., Kohler, S., Hänggi, P.: Coherence stabilization of a two-qubit gate by AC fields. Phys. Rev. Lett. 95, 140502 (2005).

    ADS  Article  Google Scholar 

  26. 26.

    Chen, P.: Geometric continuous dynamical decoupling with bounded controls. Phys. Rev. A 73, 022343 (2006).

    ADS  Article  Google Scholar 

  27. 27.

    Clausen, J., Bensky, G., Kurizki, G.: Bath-optimized minimal-energy protection of quantum operations from decoherence. Phys. Rev. Lett. 104, 040401 (2010).

    ADS  Article  Google Scholar 

  28. 28.

    Xu, X., Wang, Z., Duan, C., Huang, P., Wang, P., Wang, Y., Xu, N., Kong, X., Shi, F., Rong, X., Du, J.: Coherence-protected quantum gate by continuous dynamical decoupling in diamond. Phys. Rev. Lett. 109, 070502 (2012).

    ADS  Article  Google Scholar 

  29. 29.

    Fanchini, F.F., Hornos, J.E.M., Napolitano, RdJ: Continuously decoupling single-qubit operations from a perturbing thermal bath of scalar bosons. Phys. Rev. A 75, 022329 (2007).

    ADS  Article  Google Scholar 

  30. 30.

    Fanchini, F.F., Napolitano, RdJ: Continuous dynamical protection of two-qubit entanglement from uncorrelated dephasing, bit flipping, and dissipation. Phys. Rev. A 76, 062306 (2007).

    ADS  Article  Google Scholar 

  31. 31.

    Fanchini, F.F., Napolitano, RdJ, Çakmak, B., Caldeira, A.O.: Protecting the \(\sqrt{SWAP}\) operation from general and residual errors by continuous dynamical decoupling. Phys. Rev. A 91, 042325 (2015).

    ADS  Article  Google Scholar 

  32. 32.

    Rabl, P., Cappellaro, P., Dutt, M.V.G., Jiang, L., Maze, J.R., Lukin, M.D.: Strong magnetic coupling between an electronic spin qubit and a mechanical resonator. Phys. Rev. B 79, 041302 (2009).

    ADS  Article  Google Scholar 

  33. 33.

    Chaudhry, A.Z., Gong, J.: Decoherence control: universal protection of two-qubit states and two-qubit gates using continuous driving fields. Phys. Rev. A 85, 012315 (2012).

    ADS  Article  Google Scholar 

  34. 34.

    Cai, J., Naydenov, B., Pfeiffer, R., McGuinness, L.P., Jahnke, K.D., Jelezko, F., Plenio, M.B., Retzker, A.: Robust dynamical decoupling with concatenated continuous driving. New J. Phys. 14(11), 113023 (2012).

    ADS  MathSciNet  Article  Google Scholar 

  35. 35.

    Laraoui, A., Meriles, C.A.: Rotating frame spin dynamics of a nitrogen-vacancy center in a diamond nanocrystal. Phys. Rev. B 84, 161403 (2011).

    ADS  Article  Google Scholar 

  36. 36.

    Bermudez, A., Jelezko, F., Plenio, M.B., Retzker, A.: Electron-mediated nuclear-spin interactions between distant nitrogen-vacancy centers. Phys. Rev. Lett. 107, 150503 (2011).

    ADS  Article  Google Scholar 

  37. 37.

    Bermudez, A., Schmidt, P.O., Plenio, M.B., Retzker, A.: Robust trapped-ion quantum logic gates by continuous dynamical decoupling. Phys. Rev. A 85, 040302 (2012).

    ADS  Article  Google Scholar 

  38. 38.

    Timoney, N., Baumgart, I., Johanning, M., Varón, A., Plenio, M., Retzker, A., Wunderlich, C.: Quantum gates and memory using microwave-dressed states. Nature 476(7359), 185 (2011).

    ADS  Article  Google Scholar 

  39. 39.

    Doherty, M.W., Manson, N.B., Delaney, P., Jelezko, F., Wrachtrup, J., Hollenberg, L.C.: The nitrogen-vacancy colour centre in diamond. Phys. Rep. 528(1), 1 (2013).

    ADS  Article  Google Scholar 

  40. 40.

    Albrecht, A., Koplovitz, G., Retzker, A., Jelezko, F., Yochelis, S., Porath, D., Nevo, Y., Shoseyov, O., Paltiel, Y., Plenio, M.B.: Self-assembling hybrid diamond-biological quantum devices. New J. Phys. 16(9), 093002 (2014)

    ADS  Article  Google Scholar 

  41. 41.

    Golter, D.A., Baldwin, T.K., Wang, H.: Protecting a solid-state spin from decoherence using dressed spin states. Phys. Rev. Lett. 113, 237601 (2014).

    ADS  Article  Google Scholar 

  42. 42.

    Bacon, D., Kempe, J., Lidar, D.A., Whaley, K.B.: Universal fault-tolerant quantum computation on decoherence-free subspaces. Phys. Rev. Lett. 85, 1758 (2000).

    ADS  Article  Google Scholar 

  43. 43.

    Kempe, J., Bacon, D., Lidar, D.A., Whaley, K.B.: Theory of decoherence-free fault-tolerant universal quantum computation. Phys. Rev. A 63, 042307 (2001).

    ADS  Article  Google Scholar 

  44. 44.

    Kielpinski, D., Meyer, V., Rowe, M.A., Sackett, C.A., Itano, W.M., Monroe, C., Wineland, D.J.: A decoherence-free quantum memory using trapped ions. Science 291(5506), 1013 (2001).

    ADS  Article  Google Scholar 

  45. 45.

    Viola, L., Fortunato, E.M., Pravia, M.A., Knill, E., Laflamme, R., Cory, D.G.: Experimental realization of noiseless subsystems for quantum information processing. Science 293(5537), 2059 (2001)

    ADS  Article  Google Scholar 

  46. 46.

    Boulant, N., Pravia, M.A., Fortunato, E.M., Havel, T.F., Cory, D.G.: Experimental concatenation of quantum error correction with decoupling. Quantum Inf. Process. 1(1), 135 (2002).

    Article  Google Scholar 

  47. 47.

    Mohseni, M., Lundeen, J.S., Resch, K.J., Steinberg, A.M.: Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm. Phys. Rev. Lett. 91, 187903 (2003).

    ADS  Article  Google Scholar 

  48. 48.

    Bourennane, M., Eibl, M., Gaertner, S., Kurtsiefer, C., Cabello, A., Weinfurter, H.: Decoherence-free quantum information processing with four-photon entangled states. Phys. Rev. Lett. 92, 107901 (2004).

    ADS  Article  Google Scholar 

  49. 49.

    Altepeter, J.B., Hadley, P.G., Wendelken, S.M., Berglund, A.J., Kwiat, P.G.: Experimental investigation of a two-qubit decoherence-free subspace. Phys. Rev. Lett. 92, 147901 (2004).

    ADS  Article  Google Scholar 

  50. 50.

    Langer, C., Ozeri, R., Jost, J.D., Chiaverini, J., DeMarco, B., Ben-Kish, A., Blakestad, R.B., Britton, J., Hume, D.B., Itano, W.M., Leibfried, D., Reichle, R., Rosenband, T., Schaetz, T., Schmidt, P.O., Wineland, D.J.: Long-lived qubit memory using atomic ions. Phys. Rev. Lett. 95, 060502 (2005)

    ADS  Article  Google Scholar 

  51. 51.

    Monz, T., Kim, K., Villar, A.S., Schindler, P., Chwalla, M., Riebe, M., Roos, C.F., Häffner, H., Hänsel, W., Hennrich, M., Blatt, R.: Realization of universal ion-trap quantum computation with decoherence-free qubits. Phys. Rev. Lett. 103, 200503 (2009).

    ADS  Article  Google Scholar 

  52. 52.

    Wang, F., Huang, Y.Y., Zhang, Z.Y., Zu, C., Hou, P.Y., Yuan, X.X., Wang, W.B., Zhang, W.G., He, L., Chang, X.Y., Duan, L.M.: Room-temperature storage of quantum entanglement using decoherence-free subspace in a solid-state spin system. Phys. Rev. B 96, 134314 (2017).

    ADS  Article  Google Scholar 

  53. 53.

    Loss, D., DiVincenzo, D.P.: Quantum computation with quantum dots. Phys. Rev. A 57, 120 (1998).

    ADS  Article  Google Scholar 

  54. 54.

    Petta, J.R., Johnson, A.C., Taylor, J.M., Laird, E.A., Yacoby, A., Lukin, M.D., Marcus, C.M., Hanson, M.P., Gossard, A.C.: Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 309(5744), 2180 (2005).

    ADS  Article  Google Scholar 

  55. 55.

    Burkard, G., Loss, D., DiVincenzo, D.P.: Coupled quantum dots as quantum gates. Phys. Rev. B 59, 2070 (1999).

    ADS  Article  Google Scholar 

  56. 56.

    Usaj, G., Balseiro, C.: Design and control of spin gates in two quantum-dot arrays. Appl. Phys. Lett. 88(10), 103103 (2006)

    ADS  Article  Google Scholar 

  57. 57.

    Anderlini, M., Lee, P.J., Brown, B.L., Sebby-Strabley, J., Phillips, W.D., Porto, J.V.: Controlled exchange interaction between pairs of neutral atoms in an optical lattice. Nature 448(7152), 452 (2007).

    ADS  Article  Google Scholar 

  58. 58.

    DiVincenzo, D.P., Bacon, D., Kempe, J., Burkard, G., Whaley, K.B.: Universal quantum computation with the exchange interaction. Nature 408(6810), 339 (2000)

    ADS  Article  Google Scholar 

  59. 59.

    Makhlin, Y., Schön, G., Shnirman, A.: Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys. 73, 357 (2001).

    ADS  Article  MATH  Google Scholar 

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İ.Y. is supported by the Project RVO 68407700 and RVO 14000 and funding from the project Centre for Advanced Applied Sciences, Registry No. CZ.02.1.01/0.0/0.0/16 019/0000778, supported by the Operational Programme Research, Development and Education, co-financed by the European Structural and Investment Funds and the state budget of the Czech Republic. F.F.F. has been supported by the Brazilian agencies FAPESP under Grant No. 2017/07787-7, by CNPq under Grant No. 302280/2017-0, and by INCT-IQ. G.K. is supported by the BAGEP Award of the Science Academy and the GEBIP program of the Turkish Academy of Sciences (TUBA). G.K. is also supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under Grant No. 117F317.

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A: Derivation of the master equation

A: Derivation of the master equation

We introduce the details of the derivation of the master equation that governs the time evolution of our two-qubit system. First of all, we assume that each source of error, induced by the environment, is present separately. In other words, both independent environments for each of the qubits and the common environment, which the qubits together couple when they are close, are assumed to be present during the whole evolution. In this case, the total Hamiltonian is given by Eq. (1), with \(H_{\mathrm{int}}\) defined by Eq. (6) and \(H_{\mathrm{env}}\) defined by:

$$\begin{aligned} H_{\mathrm{env}}=\sum _{k} \omega _{k} {a_{k}}^{\dagger }a_{k}+\sum _{s=1}^{2}\sum _{k} \omega _{k}^{(s)}{a_{k}^{(s)}}^{\dagger }a_{k}^{(s)}, \end{aligned}$$

where \(\omega _{k}\) is the frequency of the kth normal mode of the environment that introduces common errors, with \(a_{k}\) and \({a_{k}}^\dagger \) being the annihilation and creation operators, respectively. In the second term, \(\omega _{k}^{(s)}\) is the frequency of the kth normal mode of the sth independent environment with respective annihilation (\(a_{k}^{(s)}\)) and creation (\({a_{k}^{(s)}}^{\dagger }\)) operators. It is important to emphasize that each error source is present independently of others since, in such a case, the total environment Hamiltonian could be written as a linear combination of the Hamiltonians of three distinct environments.

Having defined the total Hamiltonian, we utilize the Redfield master equation approach to calculate time evolution of the reduced two-qubit system:

$$\begin{aligned} \frac{{\hbox {d}}\tilde{\rho }_{S}\left( t\right) }{{\hbox {d}}t} = -\int _{0}^{t}Tr_{\mathrm{env}}\left\{ \left[ \tilde{H}_{\mathrm{int}}\left( t\right) ,\left[ \tilde{H}_{\mathrm{int}}\left( t^{\prime }\right) ,\rho _{R}\tilde{\rho }_{S}\left( t\right) \right] \right] \right\} {\hbox {d}}t^{\prime }. \end{aligned}$$


$$\begin{aligned} \tilde{H}_{\mathrm{int}}(t)=U_{\mathrm{env}}^\dagger (t) U_{\mathrm{gate}}^\dagger (t) U_{c}^\dagger (t) {H}_{\mathrm{int}}U_{c}(t) U_{\mathrm{gate}}(t) U_{\mathrm{env}}(t), \end{aligned}$$

where \(U_{\mathrm{env}}(t) = \exp (-iH_{\mathrm{env}}t)\), \(U_{\mathrm{gate}}(t)\) is the unitary evolution operator related to the time-dependent Hamiltonian \(H_{\mathrm{gate}}(t)\), given by Eq. (2), and \(U_c(t)\) is given by Eq. (9). In addition, \(\rho _E=\frac{1}{Z}\exp (-\beta H_{\mathrm{env}})\), where Z is the partition function \(Z = \mathrm{Tr_{env} }\exp (-\beta H_{\mathrm{env}})\) and \(\beta =1/k_BT\) with \(k_B\) being the Boltzmann constant, and T is the absolute temperature of the environment. Finally, \(\tilde{\rho }_{S}\left( t\right) = U_{\mathrm{gate}}^\dagger \left( t\right) U_{c}^\dagger \left( t\right) {\rho }_{S}\left( t\right) U_{c} \left( t\right) U_{\mathrm{gate}} \left( t\right) \) where \({\rho }_{S}(t)\) is the density operator in the Schrodinger picture.

To write an effective master equation in order to be solved numerically, we rewrite the interaction Hamiltonian as

$$\begin{aligned} H_{\mathrm{int}}=\sum _{n=1}^3 \varvec{\lambda }\cdot {\varvec{S}}_n(t)\otimes B_n+\varvec{\lambda ^*} \cdot {\varvec{S}}_n^{\dagger }(t) \otimes B_n^{\dagger }, \end{aligned}$$

where \(n=\{1,2,3\}\) represents each one of the three independent baths (one individual for each qubit plus a collective one for both qubits) and

$$\begin{aligned} {\varvec{S}}_1(t)= & {} \xi (t)\left[ {\varvec{\sigma }}^{(1)} + {\varvec{\sigma }}^{(2)}\right] ,\nonumber \\ {\varvec{S}}_2(t)= & {} \left[ 1-\xi (t)\right] {\varvec{\sigma }}^{(1)},\nonumber \\ {\varvec{S}}_3(t)= & {} \left[ 1-\xi (t)\right] {\varvec{\sigma }}^{(2)}, \end{aligned}$$

with \(B_n = \sum _k g_k^{n}a_k^{n}\), and \({\varvec{\lambda }} = \hat{z}\) for a dephasing environment and \({\varvec{\lambda }}=(\hat{x} + i \hat{y})\) for the amplitude damping. In the interaction picture, we can finally write the interaction Hamiltonian as:

$$\begin{aligned} \tilde{H}_{\mathrm{int}}\left( t\right) = \sum _{n=1}^3 \Lambda _n\left( t\right) \tilde{B}_n(t)+\Lambda ^{*}_n\left( t\right) \tilde{B}_n^{\dagger }\left( t\right) , \end{aligned}$$


$$\begin{aligned} \Lambda _n\left( t\right)= & {} U_{\mathrm{gate}}^\dagger \left( t\right) U_{c}^\dagger \left( t\right) \left[ {\varvec{\lambda } \cdot {\varvec{S}} }_n(t)\right] U_{c}\left( t\right) U_{\mathrm{gate}}\left( t\right) , \end{aligned}$$
$$\begin{aligned} \tilde{B}_n(t)= & {} U_{\mathrm{env}}^{\dagger }\left( t\right) (B_n) U_{\mathrm{env}}\left( t\right) . \end{aligned}$$

Replacing \(\tilde{H}_{\mathrm{int}}\left( t\right) \) in the master equation, we can write it in a more clear way:

$$\begin{aligned} \frac{\hbox {d}\tilde{\rho }_{S}\left( t\right) }{\hbox {d}t}&=\sum _{n=1}^3 \left[ \tilde{\rho }_{S}\left( t\right) \Lambda _n^{*}\left( t^{\prime }\right) ,\Lambda _n\left( t\right) \right] \mathcal{G}_{1}\left( t-t^{\prime }\right) \nonumber \\&\quad +\left[ \tilde{\rho }_{S}\left( t\right) \Lambda _n\left( t^{\prime }\right) ,\Lambda _n^{*}\left( t\right) \right] \mathcal{G}_{2}\left( t-t^{\prime }\right) +\left[ \Lambda _n^{*}\left( t\right) ,\Lambda _n\left( t^{\prime }\right) \tilde{\rho }_{S}\left( t\right) \right] \mathcal{G}_{1}^{*}\left( t-t^{\prime }\right) \!\!\nonumber \\&\quad +\left[ \Lambda _n\left( t\right) ,\Lambda _n^{*}\left( t^{\prime }\right) \tilde{\rho }_{S}\left( t\right) \right] \mathcal{G}_{2}^{*}\left( t-t^{\prime }\right) , \end{aligned}$$


$$\begin{aligned} {\mathcal G}_1(t)= & {} \int _0^\infty \hbox {d}\omega J(\omega )h(\omega )\exp (-i\omega t)\nonumber \\ {\mathcal G}_2(t)= & {} \int _0^\infty \hbox {d}\omega J(\omega )\exp (i\omega t)[h(\omega )+1], \end{aligned}$$

with \(h(\omega ) = \frac{1}{\exp (\beta \omega ) -1}\), and \(J(\omega ) = \omega \exp (-\omega /\omega _c)\) where \(\omega _c\) is the cutoff frequency. Note that to describe the environment spectrum, as usual, we assume that the number of environmental normal modes per unit frequency becomes very large. We also assume that all environments are identical since \({\mathcal G}_1(t)\) and \({\mathcal G}_2(t)\) are same for all baths. Further details can be found in Ref. [31], where the calculation has been developed for the case of time-independent interaction Hamiltonians.

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Yalçınkaya, İ., Çakmak, B., Karpat, G. et al. Continuous dynamical decoupling and decoherence-free subspaces for qubits with tunable interaction. Quantum Inf Process 18, 156 (2019).

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  • Open quantum systems
  • Dynamical decoupling
  • Quantum control
  • Quantum gates